I'm not sure about naming it because I haven't had one yet, but just take care of it as if it were your first tama. You'll be able to wed this one too and have another generation.
Well, has your mom or dad tama already left? Cause if it hasn't left than the only thing you can do to the baby is clean up it's poop, if it's sick take it to the doctor and when you feed the mom/dad it feeds its chewed up food to the baby. Is this what you were looking for? I need to know if the parent is still there to know if it is there in order to answer your question first. Hope I helped?!?
It will change to Gen2 after the parent leaves. Also, if it's your Tamas last day and you want to see it leave set the time to 11:59 and you will see the process. This is not a lie, I tried it.
Yeah, the partner always leaves. Then your parent Tama takes care of its baby until it (the parent) leaves, which will probably be tomorrow at midnight.