Under Electrum's Send tab, we can see under Outgoing payments the date/time the bitcoins were sent (2020-10-03 19:47), the amount that was sent, and the Status (which says Paid).
What matters is the "History" tab. Those "invoices" in the Send tab are only there for semantic purposes (like organizing when a payment invoice is still "valid" or has expired in the eyes of the receiver). With Bitcoin, if you send any amount to any address and at any time, the transaction will go through and the receiver will get them, regardless of any invoice (even if expired or non existent).
On the bottom of the tab, we see Synchronizing...(37/40), and it has been that way for 24 hours now. In our History tab, the money is still in our wallet.
When we reached out to the receiver, they explained they had not yet received any payment.
Was it "synchronizing" when you made the transaction? Your transaction was probably never broadcasted, thus, it was never really "sent". That would explain the receiver not seeing anything. You can double check that by looking for his address in a Bitcoin block explorer (like
https://blockstream.info) for the transaction.
Also, and I don't know if this confounds things further, in an attempt to locate our wallet address, we clicked on the Receive tab and hit the Request button (which apparently creates an on-chain request - which is jargon neither of us understand). In the "Expires after" dropdown menu, my girlfriend clicked on "1 day".
Same thing I said above for the "Send tab". Even if the "invoice" expires, it doesn't matter. The address is valid and you can send anything at any time (or receive in this case). This is basically a visual feature to control payments.
We are extraordinarily confused as to what is happening. Has the money not been sent to the receiver because it is stuck on synchronizing? If so, what is the best way for us to move forward...should we just wait? Or is our transaction simply going to expire within the next 24 hours? Also, is there any possible way that we will somehow lose the money we have attempted to send?
First of all, make sure you are on the latest version (or at least 4.0) of the wallet (and
only download it from ELECTRUM.ORG !!).
Try going to Tools -> Network -> Uncheck "Select server automatically" -> Right click another server -> "Use as server". See if it synchronizes.
If that doesn't work, try closing the wallet and reopening it again. If it still doesn't work, close it, go to the Electrum directory (on Windows, it's usually %appdata%/Electrum), deleting the "blockchain_headers" file and opening the wallet again. It will take a few seconds to synchronize and everything should work (hopefully). Then, check if the outgoing transaction you tried to make is in the History tab. If it's not and your balance is untouched, then you can just make the transaction again.