I'm reading up on upgrading. Since I changed my theme (it was 1.7) beyond all recognition and renamed it, is there going to be any theme problem once I upgrade?
That depends on how your theme is coded. There is no definitive answer. It could work fine, it could need some tweaking.
Also when upgrading, the codex doesn't mention anything about backing up database with C Panel, which is what I use. Any possible problem there?
Backing up in Cpanel is fine.
Given your first question, I suggest you also back up the entire contents of your WP root folder and all contained folders before you upgrade. The, between the two backups (files and db), you can revert to what you have now if necessary.
Last, there seems to be so many update problems here and from what I've read, no one seems to be successful using the automatic or 3 step manual process, so would it in the long run be better to simply start out with the extended update process? (I would start out with the short 3 step if not for having to backtrack and reinstall 2.7.1 if it fails).
I've successfully upgraded a number of sites from 2.7 to 2.8.4 with auto-upgrade, with no problems. However, since you are concerned, I suggest you do a manual upgrade. It only takes 5 - 10 minutes longer and avoids any possibility of the auto upgrade problems people have reported.
Once you get upgraded you should be fine in the future with auto-upgrades.