Total loss

Yes you can buy it from the insurance company and fix it. A total loss means it would cost more to fix then the vehicle is worth.

Yes but it will be a salvage title vehicle

After the insurance writes it off as a total loss they have the title, you can buy it back from them but usually it goes to a salvage auction. Not all states let members of the public buy salvage titles from insurance auctions. If yours doesn't you'll have to buy it through a broker.

If they do let you buy it back it would be easier to buy the same make and model of yours that needs what is left good on yours and transfer it to the other car that you buy cheap that has a good clear title you can use for road use as getting a salvaged title returned to a normal title for road use can be very hard and time and money consuming.

They. Are. Letting us keep the car and paying us a check

Are they keeping the title? Giving you a salvage Title instead?