You sound like a decent, caring human being, Nic. Long may you be so! In looking after Dave, don't neglect yourself.
You would not be the first person to assume that, if you are decent and caring towards someone they will reciprocate. He may of course continue to do so. But you've already noted he did not ask any questions about yourself. What's with that lack of expressed interest?
The lack of asking you anything about yourself is difficult to interpret. He's new to the job now, so he's a little on the awkward and nervous side and in need of a warm welcome.
Give him a few months and his need for your reassurance and guidance will likely reduce and he might then feel freer to distance himself from you. I am not suggesting you treat him any differently from the wonderful thoughtful way you have treated him thus far. Just look after yourself, okay?
As I said, you will not be the first person who has been decent and at some point, treated with disdain.