Below are some of my thoughts after building out my own home network over time:
1) It depends on how much it costs. I personally love having my modem, router, switches, and servers all in a centralized location where it out of sight. Not only that, it makes connecting everything together easier. If you decide later to move your ISP connection to a different location (if you rearrange your living room or remodel, etc), it is more of a hassle than if it's in a centralized location where it will likely never need to be moved again. If you want a quicker, cheaper solution that you may be able to accomplish yourself, running an Ethernet cable from the ISP or private owned modem/router to your centralized location will work as long as you don't exceed 100 meters (~328 feet).
2) If your network closet is completely enclosed, you most likely need some sort of ventilation especially if you are going to have servers in the closet. If it's just a network switch, you may not need it. PoE switches run hotter but if you don't have a heavy load on it, it might not get too hot. Some people install a fan in the top and bottom of their door to draw in cold air and blow out hot air. Others blow hot air into the next room. Since I am working to finish the basement in my house, I was able to build a 4.5 ft. x 6 ft. server room. I was fortunate in that I have a return air trunk that runs parallel to the one of the walls so I only needed to run a few feet of ductwork. I installed an AC Inifinity 6" inline duct fan which has a temperature control so it runs when it gets too warm. I am able to keep the overall temperature at 73-74 degrees or below without running the fan in its maximum power (only run it at 40% since that seems to be enough without generating too much fan noise). I have 2 servers, 3 switches, 1 router, 1 modem, a NVR, and a Raspberry Pi and it stays that cool with the fan/ventilation. I also put a passthrough vent down low on the wall near the door to allow more air to be drawn in. It looks clean and it works great. Since it's in my basement, it naturally stays cooler so that helps. In the winter, it will definitely stay plenty cool since it's on the corner of the house (I have a walkout basement).
One thing you will want to avoid is to blow the hot air outside your house. I've heard of others doing it, but when I researched it, some have said it's harder on your HVAC because it creates negative pressure. Also even though it's warm air, it is still conditioned air (still may be cooler than the outside air in the summer and in the winter, you don't want to blow the warm air outside of your home). Therefore, blowing the warm air into your return air vent works well.
3) I had to redirect some of my Ethernet/coax cable from the closet under the stairwell in the basement to the server closet I built, and I used PVC pipe inside the walls. I could barely fit all of them in the size I chose, but I didn't want to use to wide of a pipe to weaken the studs. I chose the maximum size for the maximum recommended hole you can cut into studs. I only needed to run it through a couple of studs since it was close to the corner of a wall. Then I ran more PVC conduit (used gray plastic electrical conduit) above the ceiling in the server closet to my HVAC room so I can run the wires from the closet under the stairs, across the HVAC room and into the server closet. I used two 2" conduits, and I nearly filled both of them! I wish I had ran 3" conduit but I was able to do what I needed done with 2". I don't plan on adding too much more since I tried to plan ahead and ran a bunch of drops in my basement (20 drops -- 4 at 5 different locations). The 2 floors upstairs have 16 drops in comparison (that is what I had the builder to run -- I wish I had a few more ran, but I didn't want it to cost too much).
4) If you don't have the ISP connection to your house relocated, I know that a lot of people on Reddit show off their "lack racks" that are made out of Ikea furniture or custom built out of wood. With the price of lumber these days, it's probably cheaper to buy a low end server rack. For home use, cheap racks work great (or if you can find a solid used one that a business is throwing out). Being in the living room, I understand you would want it hidden. You would want to make sure it has some ventilation since routers can get pretty hot depending on what you are using.
One thing I didn't see you mention: since you are planning to run cables and use PoE switches, you may want to consider using wireless access point(s) that are powered via PoE. Since you have a small 1 floor house, you may be able to get away with one centrally located access point. If your closet will be near the center, you won't have to run a cable very far. You could just put the access point in your closet, but it could be better if it was mounted to the ceiling near the center of your house. If your house is more rectangular, you may need 1 AP on each side of the house (or if you want to extent coverage to your yard without getting outdoor APs but range may still be limited).
