Keep in mind that ETC recently had 51% attacks and many exchanges have disabled deposits. Hence there is a reason why the profitability is better because its more riskier to mine.

Since its the same fork as ETH you can use any  ETH desktop wallet as long as you got the private key for that wallet. It will work on the ETC network also. Just do a small transaction writing a raw transaction and make sure it works... it should.

If you want to mine to an exchange... use Coinbase but keep in mind the confirmation time is like 2 weeks. Hence you just need to use your own wallet.