(Lots of) Home runs are good

You are correct that you want to run a cable to each room from the central switch. In fact, I would run at least 1 more cable than you think you will need to each room, and consider running a line or 2 to other rooms as well - especially if your walls are open. Cable is cheap, and pulling 4 cables instead of 3 is no more work when done at the same time. If you decide later that you want a 4th jack, you either need a small switch (which does limit bandwidth, not really increase latency) or you need to open walls again to pull that 4th cable.

Use a patch panel

Rather than run the cable from the big switch to each room, you should have a patch panel in between. Patch panels basically change the type of connection on the cable (the back is a 110 punch down block, front is an RJ-45 jack), and are a simple pass-through.

This is to ease installation. Pulling cable through walls is best done when the cable is un-terminated. Terminating the cable (i.e., putting the RJ-45 jacks on the end) can and is done, but punching the cable down into a patch panel is so much easier, especially for someone who has never done it before (and it sounds like neither you nor your electrician has). The cost is marginal (again, go bigger than you think you need now), but you save on headaches during installation.

You would then get keystone jacks that allow you to punch down the cable on the other end:

You shove these into wallplates on an electrical box or low-voltage plate:

They make wall-plates with different numbers of openings (usually 1-6), so you can get what you need for each room.

Finally, you would need short (1-2 ft) "patch" cables to connect the patch panel to the big switch. Buy these cables pre-made, as you won't be able to make your own for less. These are typically stranded cable, as it's more flexible.

Your final setup would look something like this:

(the top-most device with jacks is the patch panel, the middle on is the switch, and the bottom would be your router)

Buy solid copper UTP (unshielded twisted pair) cat5e or cat6 cable, rated properly (usually CMR for typical in-wall installation, but you'll need Plenum if you plan to run it in HVAC ducts), and buy multiple boxes if possible. Standard is 1000 ft but smaller lengths are available, and they come in all different colors. A decent-sized house could take 2000-3000 ft of cabling or more, depending on how many runs and where the network closet is. Again, the more boxes you have, the easier installation will be (you typically pull 1 from each box at the same time, so if you want 4 runs to a single location, having 4 boxes is easiest).

If you want things a little cleaner, you can get a wall-mounted mini rack as well:

Just make sure to get one that has the depth and vertical space (measured in "U") you need. They also make ones with hinges that make patch panel installation a bit easier.

Most product images taken from monoprice.com

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.