If there is a conflict between McAfee and Spy Sweeper, would that cause my internet connection to shut down repeatedly?

yes it can. among other things. There also seems to be a whole lot of stuff going on with mcafee. and a ton of other running processes. this in it self could kill a net connection.
So, would you shut down all of McAfee, or just the antispyware (and keep the virus scan). Or would you get rid of SpySweeper?

I've just been able to log on to the internet again after several hours of re-booting. I can almost bet that if I end the connection, I won't be able to get it back, without more of the same! Are you sure the rest of the HJT file looks OK?