An invoice is unique, so no: once paid, you cannot pay an invoice again.

Your phone should be online to receive payments, I have received payment while the app was running in the background, but never tried with the app off.

Maybe give it a try yourself. You can put money into a web wallet like coinos and coinos.io and then try to send it back after swiping away the app or whatever you want to try.

Another wallet is Zeus. They offer a lightning address which is something you can give to multiple people and they can pay as many times as they like and as much as they'd like.

They also support always running in the background mode although it could drain your battery quicker.

Finally they also support a semi-trusted offline payment solution. For this, you should open the app at least once a day to make sure payments are settled properly, and there is a fee involved for the service for each payment.