Hi,
welcome back to the Amiga users :-)
I have some answers:
1) This seems to be already a replaced battery it is not the normal one found inside of an A2000. Have a look at it from time to time and leave it inside.
2) Looks like a A2090 or A2091 controller. This is a usual A2000 controller.
3) For connecting two MFM harddisks. J0 is a SCSI-connector.
4) Yes.
5) Purchase a SCSI Cd-Rom and a SCSI harddisk. Try to obtain them via eBay, small SCSI HD's are very cheap.
6) Yes, no problem. You have to purchase a software to burn CD's like MakeCD.
7) You should purchase a Kickstart 3.1 ROM. Should be about 10,- EUR. Look at Vesalia.de or another Amiga shop.

I would prefer OS 3.1.
9) Yes you could interchange ROMs between A500 and A2000.
So far, I have to go, Noster