Everything you've mentioned are things which everyone seems to have strong personal opinions about but since you asked and I have the same car you do I'll give you my take on the questions you've asked:
As far as oil and filters go, currently I'm also running Mobil-1 5/20 in my Corolla. I have no idea what was in it before I got it; it might have been synthetic, but I doubt it. FWIW Toyota specs synthetic oil for these engines. When it's due for a change again I'll probably switch it to Amsoil synthetic. I'm running a Fram filter.
I've had good luck with Kumho Solaris KR21 tires on another car, similar to a Corolla. The PO of my car put tires on it just before I bought it, and they threw some cheap no-name's on it which I hate. When their worn-out I'll likely put the KR21's on.
I'm still running the stock battery in my car, although once in a while it seems to take a little longer to crank-over my car does tend to sit for a week or more at a time. However it sounds like you get snow where you are and colder temps tend to kill batteries. Extreme heat can as well. Where I'm at here in So.Cal. are temps are on the mild side, plus I don't have anything aftermarket installed.
Kevin