Spyware Doctor IS running on the computer. It starts automatically as shown here in the HJT log;
O23 - Service: Spyware Doctor Auxiliary Service (sdAuxService) - PC Tools - C:\Program Files\Spyware Doctor\svcntaux.exe
O23 - Service: Spyware Doctor Service (sdCoreService) - PC Tools - C:\Program Files\Spyware Doctor\swdsvc.exe

You will need to do a file search by going to Start, Search, Files and Folders and look for everything named Norton.
Delete those you find.
Then do the same thing but this time look for everything named Symantec. Delete those you find.
Yes, I would Uninstall Spyware Doctor and see if that makes a difference with accessing websites.