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Yes, I have bought all of my altcoins through Bittrex only, and I haven't had any problems till now. In fact, I like them more now after they have loaded my wallet with free Bitcoin Cash. :)

Can we also transfer buy and transfer BTC, ETC, XRP, XDG etc. to personal wallets? or we have to use the Bitrrex wallet only?

Yeah, you can transfer (or withdraw as they call it) the coins to your personal wallets any time. You can also deposit the coins from your personal wallets or other exchanges to the Bittrex wallet.

Welcome onboard! I've been using bittrex for a while now without any problem.