Not really, I mean you can put all the parts together outside of a case and get it to run, but there is no way to get them to fit in a case they aren't designed for. I always put my parts together outside of the case to check them before installing them, you just have to make sure your components aren't touching anything that'll short them. Not recommended for extended use either, just for testing
Safe way, maybe. Computer cases have a basic purpose, that's to hold your computer, keep things from interacting haphazardly with the components, and also to look nice. You could in theory do the same by putting it a plain old box, I'd assume a cardboard or wood box would work fine. Once upon a time, I had to do without a case for a little while, I just put the motherboard flat on my desk and ran it. Nothing bad happened, but obviously the danger is something shorting out or hitting the board and damaging it.
Yeah I'd usually set my mobo on the box it came with and hook everything up that way, but like I said, not something you'd want to do long term, because of heat and like navysealbrian said, keeping things from accidentally interacting with your components. Doesn't take much to damage computer parts, they are pretty fragile.