1. Your seed phrase is all you need to recover your wallet and as mentioned above by Zaguru12 above, you don't need the password when restoring your wallet from seed.
Take note that all your addresses are derived for your seed phrase and you should see the entire balance and all transactions in computer 2 unless there's transaction associated with an address generated beyond the gap limit.

The default gap limit on electrum is 20 and that means that electrum generates more and more addresses until it reaches 20 consecutive unused addresses.


2. You will get different address in the restored wallet if you use an incorrect derivation path.
In the case the seed phrase has been generated by electrum, there shouldn't be any issues with derivation path because Electrum uses a fixed derivation path and the script type is encoded in the seed phrase.
In the case the seed phrase is BIP39, it's possible that you generate the new wallet on a different derivation path and get different addresses.

3. Opening and closing the wallet is enough for the last modified time being updated.