What about "Kateřina poslala mi dlouhý dopis"?
Mi has to be the second word.
Muže být "kateřina poslala mně dlouhý dopis
No, it sounds strange. Although mně is not a strict clitic it still comes either in the second position or in the strong (first or final) positions.
I think yes. It should be accepted. I reported on that
Please check for typos before any reporting ("kateřia poslala mně dolhý dopis")