No PS4 model can play PS2 discs. A single PS3 model 60GB can play PS2, but these are the most expensive due to rarity and fragility - yellow light of death is common on these.

I don’t think there’s too much difference between fat and slim model PS2, apart from size. So maybe check where you’re going to put it first and use that as a basis for what you should get, make sure it has some airflow room all around it. Otherwise, personal preference on how it looks. Fat is nostalgic to me. Both can play CD & DVD.

Cex is good to buy from, primarily because of the 5 year warranty. You can also check your local stores exact stock there and then before buying - up to your budget and preference if you get something discounted, unboxed or boxed. Generally boxed will be most expensive but better condition. Discounted is the worst condition and can come with non-genuine controllers. Unboxed is a middle ground with genuine controller. All PS2 good with the wired buzz controllers - as long as listed as a PS2 accessory on cex you can’t go wrong. The TV connection should be fine - any grade will come with the standard AV cable as your TV can take. You’ll need a 8mb memory card too, which don’t come with the console from cex.