Hey Benjamin, welcome to the forum!
I've got the same problem of noise playing whatever position the switch is on, intermittently though. i'm planning to bring it to the Moog doctor here in France.
About the note still soudning, I remember reading a topic dealing with that lately on the forum. I don't want to say anything incorrect so I'll advise you to find this topic. Some experts gave their opinion there
At least, the third point is ok: filter can self oscillate, meaning that it can produce a note even when not fed with any signal. This occurs when resonnace is high indeed. Provided the filter cut off frequency is following the keyborad, yes, you can "play the filter".