I am a bit confused on what are the important parts to save for your wallet
If you only have your seed left (the words that is, written down on a paper or something) can you restore your whole wallet as new if everything else where lost?
Yes, provided you hadn't imported any other private keys that weren't generated from the seed. Also, just the seed wouldn't give you any labels you'd made for addresses, or record any addresses that you'd frozen or prioritised.
Do you still need to remember your password?
What about the wallet file?
What of these things are critical to remember or keep?
No, no and neither, as long as you have the key.
Are there recorded cases of theft from secured wallets? hackers getting in?
No idea about this. You'd want to be careful about malware on your machine, e.g. keyloggers.
