You're planning it incorrectly. Generally ALL jacks need a single run back to the basementUUtilities/Comms area of home where the large main switch is. The internet feed should also enter and terminate there. Then devices are connected or patched to wherever in the home you need.

Wiring 1 room to another is the wrong way to do it.

DON'T use network cable splitters, use at minimum a Gigabit switch with enough ports to do what you need, ideally with extras

Use proper WAP's designed to do the job not routers designed to do a different job. But yes you're correct they're working as designed you want them in AP mode. Many newer have that option but tons of older don't.You also want to do a Wi-Fi site survey and see what channels everything is on and spread channels out.

Checkout the pinned comments for tips on planning and layout Home Networking Basics https://www.youtube.com/watch?v=fjRKID2ucPY&list=PLqkmlrpDHy5M8Kx7zDxsSAWetAcHWtWFl

Best option and highest bandwidth is gonna be run conduit and a fiber cable. Then in garage can either terminate into a switch or media converter. The add a WAP or just plug in devices

Run exterior 1 1/2" conduit with long sweep 90⁰ bends and LB into the structure. Here's an overview

There's tons of options for hiding cables. Special trim that's hollow or has a groove in the back etc. Or simply run the cables in corner of ceiling then a piece of crown moulding to cover it etc.

The Unifi system requires a controller. Either a cloudkey, a Dream Machine type device OR the controller software running on a device on the network. But it technically only needs to run for setup or datalogging. Then can be turned off if desired.

If you have a combo box you'll have a harder time. There's very few available for that. Need to find out what modem is needed and set that up. Then use any router you want. Or find out which 3rd party combo boxes are compatible

Run some 1 1/2" conduit and run some outdoor rated fiber cable from like fs.com etc

This is a conduit overview Outdoor Conduit HowTo https://www.youtube.com/watch?v=6vmX-hona2o&list=PLqkmlrpDHy5M8Kx7zDxsSAWetAcHWtWFl&index=2

Yep you need your 2nd router set as AP mode and wired to your Router 1 not the main router.

Or if you had a 2nd run from main router to your 1st. They could have a coupler/pach cord between the 2 at the main router. I always recommend 3 runs to each location.

Yes generally it won't be an issue. But can only have 1 uplink to router or another switch so everything downstream is limited by that port throughput. Generally best to by a single large switch and connect everything to it.

If you have the ONT white box on the wall you which the router plugs in with an RJ45 cord. Then you can use any router. But if the fiber cable plugs directly into the router and it's a combo then you need the exact one. Maybe you can take old in and swap them out