The cream whipper relies on gas expansion to work.
When you make whipped cream by beating, you beat fine air bubbles into the cream. The cream traps air and becomes essentially a matrix that holds those bubbles--a foam.
Your gas-charged whipper does the same thing in a totally different way.
When you charge the whipper with gas, there's high gas pressure inside with the cream. The cream will actually absorb the nitrous oxide you put in. Because of the pressure, the gas absorbed can be thought of as really really really small bubbles within the cream. So you have a matrix of gas and cream, but because the bubbles are so small, it's essentially just cream.
Chilled liquids more easily absorb gases at high pressure, which is why it's good to use cold cream and keep the whole unit in the fridge. A limited amount of agitation (shaking) exposes more cream to the gas, improving absorption.
When you release the cream from the device, the absorbed gas expands rapidly. The bubbles get bigger, and your cream to bubble ratio becomes more like the foam that we know as whipped cream. It's really exactly the same thing, only with nitrous oxide instead of plain ol' boring air inside the bubbles.
Why nitrous oxide? As I understand it, it's because it's the cheapest non-toxic, odorless and tasteless gas you can get. Carbon dioxide would almost be a good choice, but unfortunately it's bitter. Not a good match for cream.
Finally, why is shaking too much a bad thing? That one I don't know for sure, but I know what happens when you over-whip cream with the mixer. You make butter. Perhaps the gas or high pressure encourages this conversion, or maybe you're just churning it that much when you over-shake. Either way, I'm sure you've essentially made butter when you shook it too much.
It depends on the preparation.
I have the Creative Whip which comes with a little recipe book and not all of the recipes demand that you chill after charging. For example the "instant chocolate mousse" is really instant. The only liquid ingredient is the cream itself which is already chilled, and the gas expansion inside the canister chills it even further, so it's ready right away.
On the other hand, if you're making something with gelatin, carrageenan, or any other gelling agent which needs to set, then you'll obviously need to fill the canister while it's still a liquid (otherwise it's going to be rather hard to whip, not to mention hard to get into the canister itself!) and then chill to set it.
Basically it's simply a question of how cold your liquid is before it goes into the canister vs. how cold you want it to be when you serve it. If you can chill it to the correct temperature beforehand, and that won't prevent you from whipping it (as it would with an actual gel), then by all means chill first.
In the specific case when you're using creams and/or cheeses to make a mousse or espuma, you almost never need to chill it in the whipper itself - although it won't hurt either.
By the way, if you're making a gel or anything else that needs to set, it's best to chill it gradually in an ice water bath and shake it frequently as it chills, in order to keep the gas dispersed. If you just charge it and stick it in the fridge, you may find it very hard to dispense afterward.
Best Answer
A foam is just a liquid with plenty of air incorporated into it. You can incorporate air into any liquid; in order to be able to create an actual foam, however, you need to be able to incorporate the air faster than it escapes.
What makes a liquid able to hold the air you're incorporating (and hence form a foam) is a foam stabilizer, also commonly called an emulsifier1. I know of no specific taxonomy of stabilizers, but the vast majority are hydrocolloids AKA gelling agents and belong to some family of protein.
And so on. Really almost any emulsifier will do. Basically everything in your list either is, or contains, one of the the additives mentioned above:
...you get the picture, I hope. The most basic answer I can give to this is that if you want to make a foam, you need to either use something that's already an emulsion (milk, butter, chocolate, etc.) or use an emulsifier/stabilizer additive (such as gelatin, lecithin, etc.)
If you want a relatively complete list of all of the food additives that qualify, you'll want to look at the E number, and specifically E400-499 (thickeners, stabilizers, emulsifiers).
1. As commenter Erik very correctly points out, an emulsifier is not the same thing as a foam stabilizer. However, by convention, the terms seem to be used interchangeably all over the place, to the extent that I get blank looks when I refer to a "stabilizer" as opposed to "emulsifier". So, know the difference, but don't get too hung up on it.