Dunno if this will help, but i went thro this last spring, ended up with a Spektrum DX3C transmitter, which comes with an SR300 receiver. It does everything i want it to quite well, and there are now a couple of 2 channel versions which look good. (DX2E i think) Might be cheaper.
Have finally settled on hobbyking.com nanotech batteries of various capacity and format (saddle/shorty). Cheap and seem to be reliable. Paired with a core RC50 charger, which again, does everything i ask of it.
Servos, can't help you there, i have a wide range of broken and not broken ones. But look for metal gear as a minimum. £30-40 should get you into "ok" territory. Spend more, usually get faster or stronger, until you spend lots, then they are faster AND stronger!
(FWIW, Savöx servos don't play nicely with Spektrum receivers sometimes, so maybe avoid that combination)
Pinion size will be dependant on what motor you end up with, as 5.5 is probably too fast for most outside!! I've settled on (sort of) 8.5 outdoors and 10.5 indoors. Give or take. If you get some hobbyking batteries, they do a wide selection of motors too, and pinions. I think i have 19-26 pinions in my toolbox now, on a similar car to your 22. But am still playing around, trying to decide which ones are best
But it is worth going to a club to see what everyone else running TLR 22s has for kit, will help with set ups and so on as well.