Bittrex claim they are going to credit BCC to accounts holding BTC on their exchange:
If you have a Bitcoin (BTC) balance on Bittrex during the BCC UAHF time on August 1st, 5:20am PT (12:20pm UTC), you will be additionally credited the equivalent amount of Bitcoin Cash (BCC) on a 1:1 basis. i.e. 1 BTC on Bittrex held during the on-exchange snapshot will get you 1 BCC. BTC held on orders will also be credited. Only the BTC held on your account is eligible for BCC. Full details below.
https://support.bittrex.com/hc/en-us/articles/115000808991-Statement-on-Bitcoin-Cash-BCC
Yes Bittrex should be fine too. Just not sure you can deposit new BTC there right now.
Yeah that's right, transactions in and out have been suspended for the time being.