1) I would suggest a ring. First of all, ring is one of the oldest forms of jewelry and by itself doesn’t mean much, so Asterion shouldn’t perceive it as too intrusive or insensitivity personal, especially if his relationship with PC still a bit rocky. That said, rings hold a lot of symbolism that can be used as their relationship progresses. There is signet ring, that signifies authority and transfer of authority (PC is the Master of the Hotel, so by giving ring to Asterion, he would grant him sort of blessing to freely use and act upon this authority for the betterment of the Hotel). Then it can be viewed as a magical protection ring, a kind of promise to protect minotaur and keep him out of harm’s way. And of course, the most popular ring, the engagement one, a promise of love and fidelity, although PC and Asterion are clearly not there yet, it can in time evolve to be an engagement ring. Cons: ring would probably clash with biker gloves.
2) Not when I played first time – the very thought of going into Velley put him in distress, so I didn’t force him do anything, that he didn’t want to. But eventually, on third of fourth playthough, when I was exploring all outcomes, I did send Asterion there, although at time I didn’t expect anything too serious to happen. So, I suppose, it’s a mixture of all of four options with “have been curious to know what would happen” and “have realized something bad could happen, but figured sending Asterion out was ultimately for the best because it would get them the mirror with no strings attached” being most prominent.
3) Personally I would like to give him a break. I mean, a huge pile of new people, information and events been dumped on his head and he probably needs some breathing room and time to process it all. That said, I would also like him come out of the closet. Literally. He currently resided in over-glorified pantry and he needs proper bedroom with proper bed at least until he moves in with PC. Plus, it will give him some personal space.