Bob, as much as I would like to see you switch to XenForo, I would feel much more comfortable finding the cause first.
What you described in the first post, happened to my vB site a few years ago. The problem was the server my site was hosted on was compromised. After changing all server/site related passwords, I checked each common vB file (showthread.php, index.php, etc.), and a small script was inserted at the bottom of each file. I removed the script from about (IIRC) eight different files and that fixed it. Shortly afterwards I switched hosts, 
Even though you have re-uploaded each file, and as it only affects your forum directory it still appears that the problem lies within the DB, or the files themselves.
Have you tried installing a new instance of vB in a new directory, complete with a new DB (do not wipe the old one, just create a new one)? I would try that first and see if you can access the freshly installed forum. If you can not, then either you are passing the virus on from your PC as you upload, or your server contains the malicious code. If you can access the new forum, then point the newly installed forum to the original DB. If the problem re-occurs then you know your DB is infected.
