there really is not any need for any type of "gaming chair".
your best bet is to go down to your local office supply store and sit in and play with a few chairs.
what is most important is comfort as well as any functions you listed like height adjustability, whether you can use arm supports or not, etcetera.
i've heard of all sorts of custom options such as having car racing seats mounted on a swivel base and using it as a gaming chair but really unless you want to show off or have money to blow its a waste of time.
comfortable. sturdy. options.
it doesnt have to be pretty or expensive.