You can try to re-jailbreak it. Again, make sure you're using the latest version of RedSn0w. If that still doesn't work, you'll most likely have to use the SSH method to get Cydia in there.
Yes. But to jailbreak, follow the same steps. But normally the version on your iPhone has to match the IPSW that you select. If you're jailbreaking 6.0.1, you still use 6.0. The reason is that iOS 6.0.1 is a hotfix, not a fully "patched" ios