If I didn't send MRO to polo with Payment ID, How can I get the money back
Just email them with the exact amount they seem to be sorting those sorts of issues out fairly easily, however it will be a lot more easier to claim if you didnt send a round number like 10 that someone else may have sent without a payment ID as well.
Hmmm, I have my MRO at the Cryptonote exchange and I'd really like to move it to Poloniex for the 2FA but I don't want to install the client. Should I just send it without a payment ID and do the above to have Poloniex retrieve it?
God no. It's like shooting it onto the abyss. Plus the exchange would not appreciate this strategy either IMO. The client is no hassle to install. I will transfer it for you if you like, I will not scam and wouldn't mind building some trust. But I would just install the client if I were you.