As mentioned, this is likely the post before register feature. What happens here is that users can post before they register, and then get a chance to complete a registration for them to show. If thats not completed within a short period, the data would be removed. The idea of this function is to encourage signup on your community. Of course registration still goes through the usual spam measures, so spammers would struggle to register, and their data would be deleted.
I would suggest contacting the developer on that. However this would not be in any way related to the above.
I would suggest when you are seeing this, you disable all 3rd party applications. It may well be a task that is an issue, something that is having problems running. For example, you mention you have problems with at least one item hitting an internal server error. if that is trying over and over, and possibly hitting some limit on your server, that could be causing it. This in turn could be either an issue with that item, or an issue with your server setup.
It does need to be noted here that the only way anything at all can possibly run processes which are secured by your server in some way, would be if that server is not properly secured for running them, or they are indeed being run with the credentials necessary. So this is a setup issue rather than one of the software. Its very likely, if you are running your tasks on cron, that it's the cron job that has the permissions you mention. That would not be running under the normal web user on some environments.