Yes, by using the HM01.
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It doesn't learn Cut by level, though it can learn it through the Cut HM.
Your Meowth will evolve into Persian when it reaches the Level 28. Remember that only Meowth can learn the move "Pay Day", so wait until it learn this before evolving.
Route 6.
Level 24.
Meowth can't learn and therefore not use surf because it's not a water type.