Is it possible to replicate system databases? I only ask because I had an issue previously (described here) that appeared to be impossible but has come back up again this week.
This time TempDB log was growing quickly and according to log_reuse_wait_desc, replication was enabled. Once replication was disabled the log quickly cleared. We have had meetings in our SQL group, the only people with access and I assume knowledge on how to set up replication, and no one has tried to set up replication on any databases on these brand new servers. Once again, I don't think SharePoint databases can be replicated, same for System Databases, yet we are continuing to have issues with this.
Has anyone had any experiences like this? Features turning on or off in seemingly impossible fashion?
per your first question, replication of system databases isn't supported.
Regarding your core issue:
Have you traced events that would tell you definitively that this is not being enabled by end-users? If this happens periodically, you may want to audit/trace for related events to make sure that this is indeed the case. The most common reason behind this is that indeed someone has enabled this functionality at some point (sometimes as innocent as just experimenting with the Replication wizard in SSMS).
With that said - there was a rare case I saw where running DBCC CHECKDB \ DBCC CHECKTABLE with data loss marked the database for replication during the fix (even though it wasn't a publication). Rare situation, so I think you should validate the more common one first (unless you've been doing a DBCC CHECKDB\CHECKTABLE with data loss recently).