2 different problems, in my opinion.
First one is due to engine by the sound of it, but whether an injector service would make a blind bit of difference, I know not. I'd have suggested checking the coil pack,which tends to throw up rough running once it's been working a while.
The 2 lights, snow & sport, coming on together is detailed in the Owner's Handbook. Being a (slightly) smart gearbox, it has various sensors for assessing the transmission fluid. That warning is telling you that the box needs servicing, and probably a fluid change.