If you know your wallet password, you should be able to extract the seed from the wallet by simply using the "Wallet -> Seed" menu item.
If your wallet is password protected, it'll prompt for the password, then show you the seed... if it isn't password protected, it'll just show you the seed.
Just wondering if there’s anything I can do to:
1. Synchronize on this wallet
You'll definitely need to upgrade to be able to sync. However, you *should* be able to upgrade without knowing the seed... simply make a backup of your wallet file and then install the latest version. It shouldn't overwrite the file, and the should just open your old wallet file for you.
2. Check to see if the transaction went through
Check the address on a block explorer (like blockchain.com or blockcypher)
3. Recover BTC if I’m unable synchronize/update without the seed
As mentioned, upgrading shouldn't wipe your wallet (make a backup anyway)... however, if all else fails you should be able to extract your private keys (as long as you know the wallet password)... and you can then import/sweep them into a different wallet.
In any case, I'd recommend that if, for whatever reason, you are unable to recover your seed, that you move those coins to a new Electrum wallet ASAP. That seed is your guaranteed recovery tool! Without it, you're asking for a world of hurt if your wallet file becomes inaccessible for whatever reason (harddrive failure, accidental deletion, file corruption etc)