I'll try removing Norton completely. Will I be able to re-install just the parts of Norton that I use?
I'm not sure if that is possible with Norton at all. You already stripped it down to what you need, but since it's a program suite, central components will always be installed and loaded. I liked Speeddisk very much (like you) but I decided to forget about it because I found it wasn't worth all the hassle. That was 1996 and Norton Utilities for Win95...
To uninstall, try a guide like
http://kb.winantivirus.com/index.php...uestion&id=534 or use "uninstall Norton Systemworks" on Google to find more instructions and even more Norton-bashing. You won't be able to reinstall without a complete uninstall.
My pc had been running very slow for a couple of weeks and it took forever to load web-sites that I normally don't have trouble with.
How much RAM has this computer? In Taskmanager, you can see how much CPU load and memory is used by each process. Is there a process showing something unusual?
I browsed through your processes again and I can't find anything suspicious for causing slowdowns except Norton. Only few additional processes are being loaded on startup on your computer. Try disabling the "Microsoft Location Finder" to check if this is responsible for some trouble: "Slow internet access" can be a completely different issue to "slow reacting/loading computer" and this program is the only one related to the internet and the only third-party thing besides Norton and Spysweeper.
If that all doesn't change anything, the problem is probably buried deeper in your system. But keep in mind that anything you try to fix it can make it worse, too. So backup important data first.