How to set up home network w/ fiber from street & unterminated Cat5e in garage
Without knowing more details, your plan is reasonable. Network equipment in the garage is not ideal, but not terrible, especially if insulated. On the other hand, if you live in a hot climate, the heat in the summer will be terrible for the gear. Cold is much less of a problem, though extreme cold would be.

One minor change is to add a patch panel between the switch and the drops to the rooms. This allows you to punch down all of your cables, rather than terminate them with jacks by hand; punching down is much easier for the novice (and the extra cost is minimal). You then run short cables (you purchase, not make) between the patch panel and switch. See my answer to a similar question for more details about using a patch panel.

Depending on how the house is laid out, and how much slack there is in the unterminated cables, you could take the whole bundle and move it somewhere inside the house. If there's little slack, you may just move them to the other side of the wall they're currently on. All my network and CATV stuff is in the master closet for this reason.

Brand recommendations are off-topic here, so I won't comment about specific routers or equipment. As far as testing the network, you should just need the gateway connected to test the fiber install. The rest is on you and can be done later. A simple network tester (example only, not a recommendation) will verify all your punchdowns are made securely, though that won't verify gigabit speeds.

As for multiple wired devices in one room, yes, a small 4-8 port unmanaged switch will work just fine.