Ok here are the facts i have gathered in the last few months regarding this problem.
1. BuddyPress versions prior to 1.6.1 had this exact same problem that is occurring on your website. The problem was caused by a coding issue that for some strange reason created malformed query strings. The exact same malformed query strings that are currently occurring on your websites.
2. I followed up with several people (15 people) over a period of over 3 months to check and see if upgrading to BuddyPress 1.6.1 fixed the old coding issue. Every single one of those people have confirmed 100% that the old coding problem was fixed on their websites by upgrading to BuddyPress 1.6.1. And that the old problem was no longer occurring because the query strings were no longer being malformed.
You are the only person i have come across that has upgraded to BuddyPress 1.6.1, but yet you are still experiencing the old problem that was occurring in older versions of BuddyPress.
So when you look at all of the facts, the problem appears to be very obvious - the version of BuddyPress that you have installed on your websites is not really BuddyPress 1.6.1. Or some combination of files, plugins, themes, etc on your particular websites is somehow continuing to cause the old BuddyPress problem.
The reason i mentioned looking at cache is because your websites are experiencing an old known BuddyPress problem in previous versions of BuddyPress that has been fixed as of BuddyPress 1.6.1. So logically cache might have something to do with the problem because cache might store old cached files. This was a long shot, but it does hold some logic. You can eliminate that cache has anything to do with this by clearing all of your cache.
If it was me personally troubleshooting this I would use this method that i have suggested to you already.
Install a new WordPress installation in another folder.
Install BuddyPress 1.6.1 and test it.
Install BPS and test that everything is working correctly.
Now one last thing that i will point out to you again is this.
You have Fatal php errors occurring on your website that are telling you that something is horribly wrong or broken with your BuddyPress installation. You have database errors that also indicate something is wrong or broken with your BuddyPress installation.
Based on all the clues, the actual errors on your website, the fact that 15 other people are not experiencing this problem with BuddyPress 1.6.1 all obviously lead to one conclusion - BuddyPress 1.6.1 is either damaged on your websites, did not install correctly, is missing files or your websites have database corruption or damage.
I cannot offer any more advice because i have already come to the obvious conclusion based on all the facts.