Do I have to pay twice if I want to install Logic Pro X on my laptop and desktop?

If you purchase Logic Pro X, you can install it as many times as you want in all your Macs for free, always that your Macs are supported. You only have to pay the first time you download it

No. Sign in with the same Apple ID on both computers' Mac App Store, and it will appear in the "Purchases" tab if you already bought it. You just pay once.