Now I currently testing it by sending a small about to another wallet. I should know within a couple hours if it worked.
You should know within a few seconds. Once a transaciton has been signed (sent) Electrum will display the transaction ID for you. On the history tab you'll see the transaction and it's progress. You can also right-click on the transaction in the history tab, and select "view on block explorer."
1. Does it matter weather or not we leave our wallets from Electrum open after downloading, or should they remain closed when not in use?
Electrum is a client that reads, displays, and communicates with the blockchain. The blockchain is the actual ledger that confirms coins and transactions. Your wallet addresses can accept funds at anytime, regardless of whether your wallet software is running or not. Once your software has signed a transaction, the blockchain handles the rest. You don't need the software to continue running for the transaction to confirm.
2. I know it recommends storing Electrum on another device not connecting on-line...But until I'm able to get another device to put in on, is it still ok to use going on and off line?
I have my "Hot" wallet installed on a PC that's always online. If you are confident your computer isn't compromised by malware or viruses, and safe from intruders, you'll be fine. If you want to set up a "cold" wallet, I suggest you use a computer that's had a fresh OS installed, and only Electrum installed and nothing else. Once updated and secure, disconnect the computer from the internet, create your cold wallet seed, copy the extended master public key so you can use it to create a watch-only wallet, then digitally shred the hard-drive.
3. Lastly, if I'm sending bitcoin to an onion site through (tor) instead, should I or do I need to do anything differently?
You don't send bitcoin to a website, you send it to a bitcoin address. Bitcoin addresses are bitcoin addresses, regardless of where they came from. Tor or clearnet have nothing to do with it.