There's a bit of drama going on with MEW. One of the co-founders left on somewhat negative terms. Whether that means anything for the private keys, or whether or not that person took private key logs with her... who knows?

I would rather that you used a core eth wallet, the official ethereum wallet from the core team. Or you can choose a trusted hardware wallet.

Different alts use different wallets... unless it's an ERC20 token, which is on the eth blockchain, so it also is stored in ethereum addresses.

Bitcoin is different altogether and can only be stored in bitcoin-compatible wallets.