All of these are possible:
The bolt is coming out.
This suggests motion: maybe you are currently turning it with a wrench or your hands and you're saying it's moving, or maybe you're pointing out that over time it is working itself loose on its own.
The bolt is sticking out.
This addresses the current condition of the bolt rather than suggesting motion. You are indicating that the bolt is not completely inserted.
The bolt is loose.
This also addresses the current condition of the bolt, but rather than talking about the degree of insertion, you are talking about the degree of tightness: saying it's "loose" suggests you can turn it with your hands, or easily with a wrench.
The last two aren't necessarily equivalent: a bolt can be fully inserted but still loose; and a bolt can be tight even when not fully inserted (for example, if it has jammed in the threads).
The bolt has loosened.
This is at best non-idiomatic, if not wrong. "Loosen" is most often used as a transitive verb: a person loosens a thing, for example. The bolt itself hasn't caused anything else to loosen. One could say instead "the bolt has been loosened" or "the bolt has become loose." This would mean that at some point the bolt has changed from being tight to being loose.