That address is just any address that you have used that has Bitcoin that you want in Electrum. If you don't have any such addresses, then exporting the private keys is not necessary as you have not used any addresses.

Yes.

Start Bitcoin Core. Go to Help > Debug Window > Console. In the text box at the bottom type
Where <address> is the address which you want the private key of. The output should be in black and be a single string that starts with a 5, K, or L. This is your private key.

DO NOT SHARE YOUR PRIVATE KEY WITH ANYONE. ANYONE WHO HAS YOUR PRIVATE KEYS WILL BE ABLE TO SPEND YOUR BITCOIN.

Do that for all of your addresses.

Then start Electrum. You can either import your private keys or sweep them. Importing means you will still use the same addresses, sweeping will simply move all of the Bitcoin to an address in the Electrum wallet. I recommend you sweep.

To sweep, create a new normal Electrum wallet. Make sure you record the wallet seed. Then when the wallet is made, go to Wallet > Private Keys > Sweep. Enter all of your private keys, one on each line, and the wallet will automatically sweep the Bitcoin into your wallet.

To import the private keys, go to create a new wallet but instead choose "restore a wallet or import keys". Click next. In the box, enter your private keys, one on each line. Follow the rest of the instructions and you will have created a new Electrum wallet with your keys imported.

If you want to continue to use Bitcoin Core, then yes, you will have to wait for it to finish.

There is no way to bypass reindexing. It is a part of how Bitcoin Core works. You can move to another wallet, but the wallet files are not compatible so you will have to export the private keys and import them elsewhere.

~Today earth, tomorrow moon~

Have patience. This is the best 1 month launch in the history of altcoins. The community is amazing and there are a ton of people working on services to make dogecoin more useful overall.

Future is bright. Have no fear. Keep having fun.