C41 is the chromogenic negative film process, RA4 is the chromogenic negative paper process. They are a matched pair, i.e. RA4 paper is designed to print C41 negatives.
Obviously that means you need two sets of chemistry. RA4 gear (paper, chems) is really cheap though and the chemistry lasts extremely well.
And no, you don't get contrast control unless you want to change the developer chemistry (add sulfite) or use partial bleaching (google for SLIMT). You used to be able to buy several fixed-grade papers with differing contrasts (eg Kodak sold their Endura range in Portra, Supra and Ultra grades) but now you can basically only buy high-contrast paper. It's annoying if you want lower contrast.