When I get a new laptop and delete my old game off of my old one, would Origin let me install my Sims 4 games again onto my new computer?

Yes of course. That is assuming you use your same Origin account on your new laptop. With Steam and Origin, you can install games on as many computers as you want. However, you can only have one computer actively running the game at a time.

If your cop of Sims 4 is a legal copy, yes, as long as you log into the same account. You shouldn't even need to delete the old copy, though you cannot play the game on both computers simultaneously.

Back up your games to the Cloud & then download them onto the new device BEFORE you delete them from the old device.

As long as you login to your Origin account, yes. Goes that way for any digital store: Origin, Steam, Epic, Microsoft, PlayStation network.