Yes, you can paint the wall covering and the cabinets!
On the cabinets, keep in mind that only the doors are wood (and possibly the frame around the door). The rest in a wood-grained vinyl laminate. The wall covering is also vinyl, and paint doesn't always like to stick well to vinyl. I've done it several times in manufactured homes, and I've found a few tips to improve your success.
You can vastly improve the adhesion by first removing the cabinet doors, handles/knobs and hinges. Then, wash everything you want to paint with a solution of TSP (Tri-Sodium Phosphate - available at any hardware/paint dept.) and water. Rinse everything well to remove any "grit", and dry with towels. Let everything air-dry overnight to make sure there's no water in joints, seams, etc. Next, prime everything that you want painted with "Kills" brand stain cover & primer. The oil-based version works best, but in an enclosed space like a trailer, the fumes will kill you in a matter of minutes. I'd use the latex version. It doesn't have much pigment, so the wood grain will show through. Finally, apply two coats of a high-quality latex interior paint - my personal favorite is Behr from Home Depot. An Eggshell, Satin, or Semi-gloss finish will be more scuff resistant, and will stand up better to years of cleaning/scrubbing.
Oh! and wait at least 2 weeks for the paint to "cure" before hanging anything (like 3M Command Strips, etc.) on the walls. FWIW, lighter colors will tend to make the space seem larger. Good luck with your project!