It's possible for Asterion to have children, in theory. The closest example is Storm, his mom was human and we don't know what his dad was but as P stated, his father and grandfather were both peacocks. We also see Khenbish's father, and I can only interpret that his other siblings are boar-like, since he stated that they needed passports (charms) as well. This might be different for each mythical, since Kota did not have parents, instead, he was brought to life through belief like other deities, such as the skinned tapir you find in The Hinterlands. It is not known whether or not Kota still has followers or he simply became mortal? Perhaps that is explained somewhere.
I'm not completely sure about Asterion's sexuality, but it could be possible if he were to conceive with a women. However, one of the hotel's contracts do not permit Asterion to develop relationships with guests. What that exactly entails in terms of intercouse, I do not know.
The devs highly encourage doing some detective work to find answers to questions like these. Although, I don't really care about finding answers that I don't know are there.