yea you can. you know those little blue lanterns in the mistlands? if you make them float by digging around them you can build on them. especially nice if you find one on top of a mountain

If the item is indestructible, it should work. I've seen videos where someone dug out underneath a black forest crypt and it floated. It permanently exists in that spot whether there's dirt under it or not. But for rocks, you run the risk of something hitting it wrong and dropping everything.

Generally construction has a support system and things will fall if not properly supported. I wouldn't build on "floating objects" (though I'm not sure what you mean by that), if they're floating then they are still connected to *something* though visually they may seem disconnected. Try it and see how it goes, with the knowledge that if that nebulous connection is broken then everything built on it will crumble. Kinda like mining an ore vein and using the trick of undermining it to have it all break at once--sometimes it looks like the whole vein is disconnected from the earthy but nothing breaks, until you hit finally hit that one spot that is still "connected" to the ground (though visually it doesn't look connected). That's probably what you have going on here, I guess.