OK, I now see you have a DSL Router/modem that is providing DNS and DHCP for you network. Based on the specs I read, that's a good box. So, there's your backbone of the network. It has four gigabit ports, so there's where you can plug a cable from the TV in. Another switch isn't necessary. Just use a normal CAT5/6 cable.
As to location, it doesn't matter where your router is located, if you can run your Ethernet cables to it without making a mess. Because of the wires, you may prefer to relocate the router. With DSL, you can plug the router into any phone jack in the house, as long as there isn't a DSL filter upstream from the jack. My router is in the garage, with the phone line coming straight to it, before it even feeds the house. I then send the line back to the junction box from the DSL filter/splitter. Put the router wherever it fits best for your cable runs. If it's convenient to keep it in the kitchen, no problem.
Cat5/6 cables can be run a maximum of roughly 300 feet without a switch between segments. That covers most homes pretty well. For my home, I have one of the DSL router ports feeding an 8 port gigabit switch, because I have too many connections for the four ports on the router. I'm going to have to get a 16 port switch one of these days.

For home use, there shouldn't be any issues chaining switches like this, but it does restrict your bandwidth between the switches if multiple devices are in one switch trying to reach the other. However, unless you are transferring very large files it shouldn't be a problem.
Don't try to use the other routers on the same network. They will end up fighting each other for control of the IP addresses, and unless you know exactly how to configure them to play nice, not worth the potential problems. If you want to use the switch for additional ports, no problem because it doesn't attempt to manage anything.
So, you have some Ethernet jacks already in the house. Do they terminate at a jack panel somewhere, or are the other ends just loose somewhere? They need to be plugged into a switch, either the one built into your router, or the 2nd switch. This switch must then be plugged into the router.
Once all cables are plugged in, the router will assign the necessary IP addresses and everything should work. Have at it!

Dan
LG NANO85 4K TV, Samsung JU7100 4K TV, Sony BDP-S3500, Sharp 4K Roku TV, Insignia Roku TV, Roku Ultra, Premiere and Stick, Nvidia Shield, Yamaha RX-V583 AVR.
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