You can't really fix the chip in the slate flooring.
Those scratches on the slate... wipe them down with a damp sponge? Do the scratches disappear when they're wet? If so, you can remove the scratches by simply using an artist's paint brush to paint them with clear acrylic nail polish. Post again if you want to know the science behind why this would work.
Basically, vinyl flooring is any kind of flooring made from vinyl, whether it be square tiles, or sheet goods. If your flooring consists of rectangles or other uncommon shapes made from a resilient vinyl material, it's something I've never come across. There really is a lot of overlap between "vinyl" flooring and "linoleum". Every "linoleum" (except REAL linoleum, which is made from a slurry of linseed oil, clay and cork) will have a vinyl wear layer on top of a paper backing, and so qualifies as being called "vinyl flooring". Some sheet vinyl flooring has both a vinyl wear layer and a sponge vinyl backing, and is therefore also a "vinyl flooring", but is typically called "sheet vinyl" to differentiate it from "linoleum" which typically has a paper backing. You really can't tell from looking at a vinyl flooring whether it has a paper backing or a sponge vinyl backing.
Yes, you can install Peel & Stick over what you have. If there is an embossed pattern on the flooring you have now, you should "float" the floor with a cement based floor leveller like Mapei "Planipatch". "Floating" a floor means mixing up a thin slurry of floor leveling cement and spreading that over the floor with a standard 5 inch by 11 inch plastering trowel to fill in that embossed pattern so that it doesn't "telegraph" through and show as a "ghost pattern" on the new flooring. Basically, what you do is:
1. Mix your Planipatch powder with the Planipatch Plus additive (pronounced "adhesive") and spread that over your old flooring.
2. When dry, set a bright light on the floor to exagerate the roughness of the floor and make every imperfection stand out. Scrape down any trowel ridges with a paint scraper, and vaccuum up.
3. Dilute some Planipatch Plus adhesive with three parts water, and use that solution to mix the Planipatch powder and apply another coat of floor leveler over the floor.
4. Go over the floor with a bright light, using a paint scraper to scrape off anything sticking up, and circling any depressions with chaulk. Vaccuum up.
5. Now, fill in the depressions only with more floor leveler using the Planipatch Plus additive diluted with 3 parts water.
6. Once dry, you can install whatever flooring you want over that cement film.
Another option would be to nail down thin 1/8 inch veneer plywood over the flooring you have, and then simply install your Peel & Stick over the thin veneer plywood.
And, yet another option is to pull up the underlayment in your kitchen and bathroom (with flooring still glued to it) and discard the old underlayment. Then nail down new underlayment, and install new flooring over it.
Here's how they make a house:
1. In the Beginning, there were the floor joists.
2. They nailed either 1X6 boards or 3/4 inch thick plywood over the floor joists to make a "sub-floor".
3. Then, they built walls interior made of 2X4's and exterior walls made of 2X6's over the sub-floor.
4. Then, they nailed down 1/4 to 3/8 inch thick "underlayment" to the subfloor inside each room (as defined by the 2X4 walls)
5. Then, they did everything else, including drywalling the walls, installing the windows, painting the walls and installing kitchen cabinets and counter tops, and
6. Then they installed the flooring in each room, such as carpet, sheet vinyl, cork, laminate and every other kind of flooring.
That way, if a flooring was glued down, you could always remove it by prying up the 1/4 to 3/8 inch thick underlayment in the room, and nailing down new underlayment.