I administer my server via remote solutions as well, and I see this issue with some regularity and it seems to be linked to some kind of negative interaction between auto-import and TV recording.
You could be on to something there, and with a little bit of guesswork, and at least one other condition, this may make sense.
My understanding is that when a MC Server goes to sleep it creates temporary Windows Tasks to wake the PC to record future programs. Maybe it only creates one Task, to start the next recording. Maybe it creates more, I don't know. When the Server wakes again, those temporary Tasks are deleted. If those tasks are created to run using the Windows System Account, then the MC TV recording process will run under the System Account. Hence, one copy of MC is running under the System Account.
Then if Auto-Import is triggered on the Server while it is recording, since the server is now awake and running under the User Account it was originally started in, that is a second copy of MC trying to run at the same time. Hence, a lock until the recording finishes. Or in Muzicman0's case, MC can't be launched, and reports that another copy is running.
So if the Server is allowed to sleep, and an automatically generated Windows Task wakes the Server to make a recording, then you could see the problems reported. Do you guys allow your Server to sleep? Remote managing the Server may also be a contributor to this, in the way it connects to the Server.
Also Muzicman0, would there be a regular recording on Tuesday that is started when the Server is asleep? That could explain why you have seen it on two Tuesdays.
So when you see the problem, open Task Manager and "Show processes from all users", and see if there is a MC process running under the System or other internal Account. Any Account other than the User Account the PC is logged on with...
But, I'm just guessing here.
PS: You may be able to test if the above is the problem by ticking the "Allow multiple instances to run at one time" flag in Options>General>Advanced. But that isn't a solution, as that flag doesn't persist through a restart of MC, for deliberately design reasons.