1. An armband or perhaps a ring with a Minotaur crest/insignia or some other symbol which Asterion would cherish.
2. I personally sent Asterion out to the valley because I felt like I had to in order to get the full story. I hated doing it because I knew it would hurt Asterion, but I was doing one save where I was perfectly nice to Asterion, one where I did what I would do normally with Asterion, and one where I did every negative thing I could do to Asterion so that I could get everything the game had to offer.
3. I would love it if you could tell Asterion advice related to your attribute, and as the story progressed Asterion would begin to be more proficent in your attribute's feild. For instance, if you chose the Art attribute, perhaps he could have more dialogue sprinkled in about about certain paintings in the hotel, and seem more pasionate about other artistic fields. Or if you chose Leader, then Asterion could have instances where he was especially charismatic to yourself or guests. I think it would be best if these were more subtle and scattered throughout - they would be like icing on the cake.
There's already dialogue that's attribute specific about your character throughout the game, but by slowly extending the attribute specific dialogue to include Asterion's actions as well I think it would help to further cement that you were having a direct impact on Asterion.