Running an ethernet cable to TV from airport as opposed to cable modem

If you run a few tests.......

What is the Internet speed when your computer uis connected directly to the modem?

What is the Internet speed when your computer is connected directly to the AirPort, (which is connected to the modem)?

They should be the same. If not, you'll need to start the troubleshooting process to figure out why they are different.

I would rather not connect it directly to the cable modem as it would then technically be on a separate network than all of my other devices, but if I would get better performance that way, then I will do that.

If your AirPort is set up in Bridge Mode......as it should be if you have a modem/router......then all devices are on the same network. A simple check of IP Addresses will confirm this.

It sounds like what you are calling your "modem" is really a modem/router or gateway device. A simple "modem" has only one Ethernet port. A modem/router or gateway will have multiple Ethernet ports.

That will depend on how the Airport is connected to the modem. If it's wired to the modem, then no, no slowdown should happen.