You can have up to 4 hatchlings, and up to 4 eggs, but if you have 4 hatchlings the system seems to not let you get any more eggs.
My new winter dragon also has turned out to be male. That's rather interesting (and convenient-- hopefully he and Mei Bai Yi will be interested in one another....)
BookishDreamer, once they've turned into adults, you don't need to do anything further with them for them to live. You can release them into the wild, but they lose their names if you do. Or you can see if they're interested in breeding with one another, in which case you might get more eggs.
Apparently if the adults don't get any views at all for 8 weeks, they lose their names, though.