Learn English – “Toys out of the pram” expression

This expression by nature is very personal, so you'd never see a "formal" version of this expression. This is a light-hearted way to ask what someone's hopes and dreams are. It is about as polite as you can be for this sort of question. Asking someone what their passion is, for instance, might demand a serious response whereas "What floats your boat?" is flexible and allows a non-serious answer, making it all the more polite to use than an alternative.

Though that said, here are a couple alternatives: "What drives you?" "What's your passion?" "What do you do for a living?" (not precisely the same thing, but you might get an equally in-depth answer)

The term crashing could fit. It's slang for staying somewhere, usually implied that it's staying and not paying. I don't know if this slang is common where you or your audience are, so it may or may not work. Other slang, as Unreason mentioned, would be sofa-surfing or couch-surfing.

I had to cut expenses by moving out of my place and crashing at my friends' apartments.

A more formal term could be residing, but it's not implicit (not to me, at least) that you didn't have to pay.

I don't know of any formal words that would imply staying rent-free, so you may just have to say it that (or a similar) way, maybe like this:

I had to cut expenses by moving out of my place and staying at friends' apartments, rent-free thanks to their graciousness.