Question: Can you get a decent projector roughly the size of a phone (in desk space) for a price that won't break the bank-
The answer is yes-ish. It can't be tilted (not for technical reasons, it'll just distort the picture and there's no way to correct it) and it definitely does not do close-range projection. But if you can get it the right distance from the wall and elevated properly, it throws a serviceable projection onto the wall. (Or ceiling. It's square and points up decently well.) You're not going to be marveling in awe at the realistic graphics of anything any time soon, but you paid less than $500 for this projector, so that's fair.
It comes with a remote (sans batteries), and a wall-charger. It has a built-in battery which can be enabled / disabled with a tiny little switch, which will sacrifice brightness (according to the manual) but otherwise let it run without external power. Sound quality's okay, but you can plug in an external set of speakers to improve that. It'll run a Chromecast with no problem, which can make it a portable Netflix / Youtube machine. It's portable, and feels reasonably robust. (I'm not gonna punt it down a hallway any time soon, mind)
Note that the remote is necessary if you want to fiddle with any of the few settings it has; the buttons on the machine itself can only change source (HDMI, AV in, or removable storage), and adjust volume.
It does everything it says it'll do, and given the price, at a rather decent value. If you need a projector you don't mind losing or breaking, this might be your best shot.