The questions seem to have started with how things stay together to how they stay apart.
It's quite a big subject and a lot of the answers are more about exploring what is meant when we say that two bits are part of the same object ... what do we mean when we say that two things are touching - and so on. A lot of confusion comes from being imprecise about these things - but being precise will make for too much writing. So what we do is provide a bunch of rules for which there will be some exceptions.
Very generally, seperateness and togetherness are emergent properties of underlying electromagnetic interactions.
There is a sense in which we can say that nothing is really separate from anything else ... being careful about the definition of "really". But there is also a more immediate sense in which I like to think the table I'm sitting at is not part of me and the table owner encourages this way of thinking. A lot of this thread, therefore, is in the context of this way of thinking.We can do a lot by thinking of the common understandings and investigating what happens. i.e. a smooth surface that we see is the strong scattering region for the ambient light averaged over many detection events in our eyes and processed in the visual cortex before becoming part of our conscious awareness ... the link between physics, biology, chemistry, and consciousness being an, as yet, unsolved problem.
When we say two objects are touching, then actually measure the gap very accurately, we will discover that it is not zero ... it's just small enough for the description "touching" to make sense... usually that means the gap is too small to see with the eye, or by some, otherwise useful, measuring process.The old saw is about why one object does not pass through another ... both are made of atoms, and are mostly empty space (it goes) and there is plenty of room for the atoms to pass each other, so why can't you push your hand through the table or walk through walls?
Practically all the things we think of, in every day terms, as surface properties are about electrons ... objects like your hand and the table stay separate in normal circumstances because the electrons of your hand repel the electrons in the table ... the repulsion increases as they get closer and, classically, it gets infinite for zero separation: so there is no way you can exert enough force with just your muscles to push your hand through the table.
Something like a biological organism is basically a complicated bag of stuff, with bits and bobs all tangled up.
Every atom in your body is chemically bonded to some other atom in your body ... but not all the molecules are chemically bonded to each other, there are cells floating in fluid in your veins and arteries for example. When you get beyond chemistry it's more about which bag the stuff is in or how the fibers are wrapped around each other.
Bags contain stuff for the same reason you cannot push your hand through the table.
Some stuff is stickier than others - look up the origin of friction and adhesion: it's a whole field of study by itself - but it boils down to how the electrons are arranged close to the classical surface of the stuff. The stickiness is how fibers can get wrapped around each other without having zero separation between any of their parts.
When the killer chair eats your butt, it does not have to break chemical bonds - it just has to tear the fibers apart.
In general, this may involve breaking weak bonds and adhesion and so on just because bodies are very complicated.
And so on and on - it is a very big subject.
For more in-depth you should be reading about the theory of complex systems (chaos math for example - and cellular automata).
In these forums we can handle simpler systems like two blocks of metal being rubbed together - makes for less typing ;)