I believe I stumbled on the problem with the private messages. There is an option in the Tapatalk admin page called "PM push". I disabled that and it appears that the private messages are working properly again.
It appears that Tapatalk is having some real problems. Honestly, if it causes any more problems on this site, I may have to just pull it completely.
Still no luck with that problem with creating new threads, though. I disabled ALL Tapatalk options, and it made no difference.
That 500 error means that there is possibly something in one of the files that Sinan was testing that went south. Perhaps a parameter being passed is missing or incorrect, and the server cannot run the PHP file affected. Chances of me stumbling on that are pretty slim, to be honest.
