(Not a teacher)
Methinks isn't common at all presently. If it is used, it is normally tongue-in-cheek, or a direct quote from something like Shakespeare - 'Methinks the lady doth protest too much' (which is actually a misquotation, but oh well, it's what people say!). No one uses it in a serious way.
Misdo, I've never actually heard or read that I can remember. It seems less archaic than 'methinks' though. I don't see why it wouldn't be appropriate in modern writing.
Best to wait for someone who knows a bit more about these things to reply.
Student or Learner