The difference between whipped cream and whipping cream

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.

• Agar, carrageenan, alginate, xanthan, and pectin are all types of polysaccharide;
• Lecithin is mostly a random collection of phospholipids;
• Gelatin is denatured collagen, i.e. animal protein;
• Whey protein is the prevalent protein in dairy products;

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:

• Chocolate is almost always emulsified with soy lecithin;
• Eggs contain high amounts of lecithin;
• Milk and cream contain whey protein;
• Most "supermarket cream" also has emulsifiers like carrageenan already in it.

...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.}

I don't think so. In the mayonnaise case, all that has happened is the fat has come out of emulsion and gathered together again, so you can re-emulsify it. In the whipped cream case, you've started to create large fat crystals (butter). I guess technically it might work to heat it up to melt the butter, re-emulsify it into cream and then beat it again, but I don't think it would be worth the effort. You could go ahead and finish churning it into butter and buttermilk if you want to avoid wasting the cream.