1 - Look at how a cell in a battery works. Eg. a Daniel cell. At the anode a zinc plate is put into the solution and some zinc atoms dissolve, leaving their electrons behind. At first this is easy. The Zn++ ions are attracted back by the electrons, but the water molecules are attracted to them and they are quickly surrounded by first 6, then more, distancing them and making them less attracted to the electrons on the anode. But as more Zn atoms dissolve, there are more electrons left behind and their combined attraction for the Zn++ ions becomes so strong that atoms no longer dissolve. The anode now has a negative charge.
At the cathode an opposite reaction is occurring with Cu++ ions coming out of solution onto the copper cathode. (The opposite behaviour of Cu here is related to its electron structure making it less attractive to water molecules and more to be part of solid copper. You'll have to ask Borek for better explanation of that.)
Again there is a limit to this. Once the cathode is positively charged sufficiently, it repels any further ions from depositing.
Now you have an equilibrium. Nothing more will change. The anode has excess electrons and the cathode a deficiency. The electrons can only get from one to the other by reversing the reactions That doen't happen because the reactions have reached their balance point, where it would require an input of energy to make it change.
Once you connect a wire between them, problem solved. Electrons can leave the anode and enter the cathode reducing the charge on both. But as soon as that happens, they have disturbed the equilibrium and the reactions carry on, at the rate allowed by the current. When you disconnect, the reactions stop because the equilibria are reestablished.
The potential difference between the electrodes reflects the difference in the charges on the electrodes at equilibrium with the solutions, which in turn reflects the chemical energy change of atoms going into or coming out of solution.
4- Chemical energy from the ionic reactions produces electrical PE. That causes a current which dissipates enrgy (produces heat) in the resistance of the circuit. Everything balances. Just the right amount of Zinc dissolves to maintain the electric potential at the electrodes and the current through the circuit. If you let more current flow, say adding bulbs in parallel, then the reactions go faster and more Zn dissolves.
When there is no more zinc left - or usually the remaining zinc has disintegrated and is no longer part of the electrode - it stops working.
On the way it may not work perfectly all the time. Ions can only react with the surface of the metals. If the concentrations get too upset from a lot of use, reactions slow down and limit the current or lower the PD. In some cells bubbles of gas (eg. hydrogen) form on the surface and reduce the reaction. Temperature will have an effect. As the electrode dissolves away, there may be less surface, its resistance may increase, less of the surface may be active due to exposing impurities or deposition of impurities from the electrolyte and the electrolyte composition will permanently change.
It may recover from short term heavy use when allowed time to recover (eg hydrogen bubbles to disappear, electrolyte to become more mixed), but will inevitably deteriorate from cumulative use.
3- The PD across anything connected from anode to cathode is the same. It is the difference between the electrode potentials. The internal reactions always tend to maintain that same potential.
Resistors in parallel are each connected to both electrodes directly, so each sees the same PD.
Resistors in series are not each connected directly to both electrodes, so do not each see the full PD.
Other batteries have different chemistries, but the general principle is the same. Copper and Zinc are just more familiar to me.
