Possible, yes. I would not get my hopes too high though. Is there no second card slot?

I used test disk to recover card miffed by a GoPro

https://www.cgsecurity.org/wiki/TestDisk_Download

Anything actually overwriten is gone for good. But only costs you time to try.

Yes

You need software, or pay a company to do it for you

You format a card blank and realize it: There is a good chance to recover it with recovery software.

You formatted the card and took some shots: There is some chance you can recover some photos but if another photo was written at the same location of where a file was that you want to recover (you really cannot tell where a photo was from most information that is normally presented to you, so the best way to do is try to recover and see what you get but it’s a roll of the dice, many cards will try not to write in the same location all the time to even out the use of the card so you may have some luck, but you won’t know until you try).

If you formatted the card and then shot until you filled the card with new photos: you’re most likely hosed.

You should try the software recommended.

If you have overwritten the data you’ll be out of luck. Imagine your images slash data as occupying mail box holes. When you write information onto the card, you are pushing all the old data out and are replacing it with new data. When you format a card, it tells the computer that all boxes are ready to receive information. While they are indicating that they are empty, they still have the old data in the box, ready to be pushed out.

Data recovery software looks at the boxes for data that hasn’t been pushed out yet.