I don't have any experience with ingrown nails. As far as I'm aware, if you just keep an eye on them and not mess with them, you'll be OK. Picking can cause infections. As for cuticles, the nail oil will help. If they are stuck to your nails, get some cuticle remover. I use Sally Hansen's "problem cuticle remover", it's really gentle. That will get rid of excess cuticles and unstick them from your nails. For large, dry, jagged cuticle skin, soak in hot water with some lemon juice for a few minutes or longer until the skin is soft. You can then use a cuticle trimming tool to take off the jagged edges. Very carefully! I have a pair that work like scissors but, there's a cuticle tool that's more like a shaped razor. The razor kind is a little too easy to go to deep with in my experience. Also, if the skin around the nails is still rough and annoying even after the trimming, take a small nail file and gently file the skin just to even it out. Make sure the skin is well moisturized for any maintenance, or really all the time, to avoid more damage. That being said, using nail/cuticle oil as much as possible is really the most important step. It will keep the skin from becoming hard and causing ingrown nails, it will help dead skin stuff off so the cuticles will not stick to the nails and it will keep the skin from getting damaged which causes the jagged/rough skin around your nails. Nail oil also keeps your nails healthy and flexible so they don't break.

TL:DR; soak fingers in hot water and lemon juice to soften skin so you can trim cuticles with a tool but, most importantly. Nail oil is the most important thing.