Since it's gotten colder outside, I've noticed that in the morning when the car is still cold, I have to ease into 2nd gear, or it grinds. Is this normal-ish, or is it a sign of a worn synchro?

 

Lots of manual transmissions do that. Its normal even when they are new sometimes. Things that can help that are changing out the fluid for something a bit thinner than whats in there or double clutching 2nd. My other car does this in the cold so i double clutch 2nd and 3rd until its warm. Its just a lot less stress on the syncros. As you said it is syncro related. The syncros are not moving fast enough to make your shift smooth so you get a grind if your not careful.