The kid thing might actually be answered, just not exactly in the way people seem to like to speculate about here (I have very little to say other than “I love Jean-Luc the way he is and I would discuss those details with him in private because keeping this private seems like something he would appreciate”.)
Remember that “Illuminerdi” website that leaked some character sheets for SNW and PIC? I checked their leak for PIC season 2 yesterday and EVERY SINGLE character description turned out to be accurate. (Same goes for their SNW character sheet btw.) They seem to be the real deal, at least in this regard. And they announced a son Jean-Luc didn’t know he has for season 3. IIRC they described him as a guy in his early 20s who is caught up in a galaxy-wide conspiracy and who needs to find out who he really is.
This plays in nicely with what we know so far about season 3, what with it “shaking up the Trek universe” and Sir Patrick saying that Jean-Luc is still facing his fears in season 3 (a son would sure shake up everything even MORE for him). Also, it would be another TOS movie callback (David Marcus, anyone), which is yet another theme they’ll be going for. So, like I said, the kid question might get an answer. And maybe even a reference as to whether he can now still have kids or not.