I think you have to check in later? Tasks and stuff should refresh after like 6 hours or so. It's been a while since I last played but there were morning and night tasks if I remember correctly. Or maybe you have to wait until morning tomorrow.
As for adoption it's definitely not required to progress! You're "supposed" to adopt according to the devs, but I've abandoned my town a lot and have had mayors retire with no kids. The game just makes you adopt a new peep like when you started.