You shouldn't even have any address under "Used" if you have no transactions. Can you post the address? And your address from bitcoin core too just in case.
That's not necessary but should've worked. It will stay pending until it expires unless the address receive the requested bitcoins or if you delete the request.
I have done some basic transactions in Electrum before, though never using a private key. The Electrum address is 1KpHseLmAmchBnMvVmoJ8nmoutNTEobkDW. It was actually two bitcoin core addresses, though I entered both of the private keys at the same time into Electrum. The core addresses are 1J5BUcQbYN5Z4e1rY8JbKKHZQ98rctXxvY and 1HcVG8ipPAVTdVMENfxWLXZSwUMBhG1JLE.
Do I have any options if it expires without my Electrum wallet receiving the BTC?
I see both of the bitcoin core addresses still have bitcoins. Try sweeping again and don't forget to click "broadcast" when the transaction detail appears.
The request feature are not needed to receive bitcoins. Even if a request disappears you'll still receive bitcoins if someone sends some to your address.