Route 6.
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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.
Level 24.
Meowth is not in the wild in Pokemon Yellow. You have to trade to get it
Pokemon Yellow does not have meowth and it can only be obtained by trade.
Yes, by using the HM01.