It is perfectly possible. Your only task is to secure the private keys. These are short enough to be printed as a QR code or written down. keep that piece of paper somewhere safe. Alternatively, it is possible to use a good, random, long passphrase as your de facto private key. That way, the keys to your wallet are literally stored in your mind and nowhere else.

As of now, there are no known fatal flaws or weaknesses in the Bitcoin protocol itself. The network is secure and as long as you manage to secure your private keys, Bitcoin is extremely safe.

It is entirely possible. Being a marketplace their software sends coins around for all sorts of reason, this is what was hacked. Someone's personal bit coins are not at risk.