Well this can be a subject with many levels of detail, and the right answer to your questions depends on how simplified or restricted of a context you want to conceive of the answers in.
Check out this fellow's web page(s) for some answers:
http://amasci.com/amateur/elecdir.html
In the simplified case of *CIRCUIT THEORY*, yes electricity always is considered to flow in circuits, whiich are made up of some collections of closed loops. The reason for the closed loops isn't because electricity MUST flow in a circuit (closed loop), it's because small closed loops of electronic devices (circuits) are very useful for many things, so they're often designed to function that way and not in any other!
It is like if you're playing golf or soccer or football -- if
you send the ball off to somewhere, you'll have to close the loop and come to meet it again, otherwise you will end up with no ball to continue to play with. If you continually knock the balls off your playing field (circuit) then there will come to be a shortage of balls on the field, and a build-up of balls some locations outside of the field where they've all been knocked away! Usually it's more convenient to keep playing with the same balls, so we make them follow circuit paths so that we can always re-use the same set of balls.
In electrical circuits, charges can flow in one direction around the circuit paths 'pumped' by electric fields that push negative charges in one direction, and which simultaneously push positive charges in an opposing direction. When the circuits stay closed the electrical charges are generally in balance of equal quantities inside the circuit, and it's the flow of them through the circuit components that does useful work or signal transfer.
In DC (direct current) circuits the electric pumping fields cause the negative charges to move in one direction and the positive charges to move in an opposing direction, and at any specific place the polarity of the electric charge current flow and the electric pumping fields that cause the flow remains the same.
In AC electric circuits the pumping electic field polarity changes frequently, so sometimes charges of a given polarity flow in one direction, and when the pumping field polarity changes to the opposite value, the charges of a given polarity flow in the opposite of the previous direction.
In wires, wire coils, capacitors and resistors and
similar simple electric components made of conductive substances like metal, many of the negatively charged electrons are mobile and so free to flow around in the conductor. There are also fairly fixed lattice works of other non-mobile electrons and atomic nuclei that tend to stay in solid positions. So for common metallic circuits the current is basically a flow of the mobile negative electrons.
In common electron (vacuum) tubes a special hot filiament or special cathode material emits some free mobile electrons, and these can flow around inside the tube according to the electic and magnetic fields that attract or repel the free electrons in the tube.
In common electron tubes that are designed to act as circuit elements, there are metallic terminal connections to structures that act as current input, current output,
and electric or magnetic field control points.
So even though inside the tube the electrons are not confined inside a wire, there is an overall flow of electrons around specific paths composing an electric circuit, it's just that parts of such circuit paths can sometimes be in metal, sometimes in vacuum, or sometimes in other materials.
When electricity flows through a semiconductor it generally
follows specific paths, and the flow is made of some mobile negative electrons, and also positive places that are 'holes' absent of electrons that would normally be there. Generally the flow of charge is still in and out of specific terminals on the device that are interconnected as points of an overall circuit.
When electric current flows through something like biological tissue or a conductive chemical solution in general (which is called an electrolyte), the current is carried by various negative species of chemical ions, various positive species of chemical ions, and possibly also some electrons as well. The presence of the electric current can have the action of causing chemical reactions between the ions and other ions or electrons. It's also common that the chemical reactions between ions, conductors, electrons can CAUSE electic flow which is what a battery cell is. In photosynthesis a photosynthetic center of plant or bacteria can accept a photon of light and use that energy to free an electron from a chemical compound and create ions and electron flow within the chemicals to provide energy for the organism; so light can act as a pump to create an electric current or separation of electric charges where there was none before.
Similarly in some organisms chemical ionic/electronic processes can create light from the electronic and chemical energy they have within them. Similarly LEDs, LASERs, light bulbs, some kinds of vacuum tubes, radio transmitters, et. al. can convert electric current or electric energy into light or radio waves which then can leave the area of the circuit.
If the electric current is of sufficiently high frequency the concept of 'circuit' paths can tend to top being completely useful in describing the interaction of electric current, electric charges, and electromagnetic fields with things. An antenna, for instance, can be just a straight wire that is only connected to an electric circuit at one end, but nevertheless electric currents and fields can flow in and around the antenna in complex ways even though there's no physical conductor connection to the 'other side' of the antenna to make any 'circuit'! Indeed electric power goes in one place into an antenna and out of the antenna comes electromagnetic waves that can travel forever away from the antenna and so the energy may never return to the circuit. The same can be true of a light bulb, etc.
In a device like an electrostatic ionizer, or in more complicated kinds of devices like the electron 'guns' in vacuum tubes or ion 'guns' in scientific equipment, heat, light, chemistry, or electric power is applied to cause the production of 'free' electrons or 'free' ions that can be accellerated and sent physically away from the circuit. Maybe these free electrons or ions are intended to cause cause distant chemical changes, maybe they are intended to convey power or information over a distance, but in any case it's possible to have such electric currents be sent 'free' of the circuit that produced them and used to interact with other materials that are relatively or totally isolated from the circuit that produced them. This is like the case of knocking the ball out of your play arena; you CAN do it, but you end up having to supply a continual source of new balls and power to send them off because you may not be getting the power or charge(s) back along your circuit elements in any direct or prompt fashion!
To use physical examples, flames, explosions, lightning, or matter jets from astronomical objects like black holes or pulsars / neutron stars can send away quite a lot of electrically charged currents off into space as electrons, ions, light, radio waves, X-rays, gamma rays, cosmic rays, etc. and the energy does not return in a circuit.
Overall the universe is pretty electrically 'neutral' in the sense that there is a very very very very well balanced amount of positive electric charge in all the protons and negative electric charge in all the electrons overall. And overall in a universal sense it's mostly impossible to alter
the balance of positive and negative charges, or to create new positive charge without also creating new negative charge, or to destroy/consume positive charge without also the destruction/consumption of negative charge. So although there's no requirement that any PARTICULAR charge or electic current stay confined to any manmade or natural electrical or chemical / ionic circuit or place, in general things must stay relatively balanced in terms of the positive and negative charges present in any particular region of space, otherwise nature will tend to cause the rebalancing of the charges by having charges become attracted from elsewhere to provide balance in all locations.
So when you talk about 'earthing' a circuit you're just implying that the circuit is connected to the Earth which is a sort of giant reference point and reservoir for electric potential and charge flows so that everything on the surface of the Earth stays electrically at more-or-less the same electrical potential, and currents that leave the Earth in one place will tend to (at least) return to or THROUGH the Earth in some other place.
Of course in the form of the solar wind and cosmic rays the planet Earth itself is always being bombarded with currents of ionic and electronic and muonic charged particles all the time from elsewhere in the solar system and universe.
Similarly the natural action of electric pumping causes massive amounts of electric charges to be pumped from the ground up into the sky / clouds until ultimately thousands of times every second lightning bolts (and ionic winds) cause the imbalances of charges to equalize closer to equilibrium over space...
Also it's important to understand what you mean to track
in the flow of things.
Electric charge can be an electron, 'hole', ion, proton, muon, or other kinds of elementary charged particles.
You generally don't create or destroy new elementary
charged particles (in any great quantity), but you can
create ions or 'holes' by moving electrons around from
place to place. Like the ball playing analogy, you're
free to use energy to knock any of these charges about as much as you wish to, even clear out of the 'playing arena' of your circuit, but if you don't allow balancing currents of other charges to flow in from elsewhere, you'll end up with a great unbalanced electrostatic charge and perpahs the inability to summon enough energy to keep flinging
charges away from your circuit as it becomes more and more electrostatically charged from all the cast-off charges.
Electric current is just the motion / flow of electric charge,
so the ability of current to stay in your circuit (or not) is the same as saying whether charges stay in your circuit. If you drop a television set into a swimming pool while it's on, the charges most certainly won't stay confined within the circuit, though in such a case a new circuit will be created between the AC electric mains, through the water, and through the Earth itself for electric current to flow through!
Electric energy can actually be either electric potential energy, the kinetic energy of moving charges, or magnetic potential energy, or energy in the form of an electromagnetic wave.
The electric potential energy is often stored in circuit elements called capacitors, though everything in the universe has capacitive electric potential energy relative to every other object, so one could refer to that which is associated strongly with one's circuit, or
one can choose to be more general. Also in capacitors the electric field energy is actually stored in the insulator space BETWEEN the plates whether that's vacuum, air, plastic, etc. And true currents of electric charges can't flow through an insulator, so true currents of charges can't flow 'through' a capacitor even though we say that a capacitor has current flowing through it in an AC sense; in that case the current doesn't REALLY flow THROUGH it, so much as it is influenced to flow through the circuit by the potential of the energy stored in the capacitor...
Magnetic potential energy is usually stored by inductor
circuit components, but actually the true energy is generally mostly in the space AROUND the circuit wires (e.g. in the magnetic material core near the inductor or the very air/empty space AROUND the wire)! So it's really
a simplification to say that it's 'in' the circuit, actually what's being said is that the circuit is designed to efficiently create and use such energy which doesn't really reside totally 'in' the circuit.
Electromagnetic wave energy often doesn't stay in
or around a circuit at all, as in radio waves, light, x-rays,
etc. Circuits can help produce or consume such energy,
but it often travels independently of the circuit.
The fundamental physics that controls where electric,
magnetic, electromagnetic energy flows is electrodynamics
and it's much more comprehensive than circuit
theory which is just a very simplified child of it that
ignores all cases except where circuit components
of inductors, capacitors, resistors, batteries, and active
devices like some transistors / ICs / tubes.
When talking about electric flow in electrolytes, chemical
solutions, tissues, molecules, that's really better explained
by quantum chemistry, electrochemistry theory,
biochemistry, etc.
When talking about charged particle electric flow in
space near the Earth that's more well covered by
meteorology, theories about geomagnetism, ionospheric
physics, et. al.
Radio engineering references will tell you more about
the ways electricity flows around antennas, resonators,
through the Earth at high frequencies, through the air
and ionosphere, etc.
Of course there are also disciplines of engineering that
study electron tubes and charged particle devices like
electron/ion sources; you're right in saying that these
don't categorically have to behave strictly as
circuit components in the most simple classical sense.
One wouldn't have too much luck using basic circuit theory
only to explain a cathode ray tube or x-ray tube or klystron.
