I'd leave it (texture is pretty common where I live so it doesn't bother me a bit) but paint it a bit glossier.

Otherwise, have it smoothed out.

You should address (or look under to see if you need to address) the light fixture before you change the ceiling texture. If it requires a big patch, you want to do that first.

Love the tile.

Peel and stick, of anything, including wallpaper, or even wallpaper you paste, is not going to stick with that uneven texture ceiling. However, like Fori, it is very common type ceiling in my area. My house now, has smooth, but, my last house, had that in every room. I would give it a good coat of primer, and 2 coats of flat white paint, and it will help it recede a bit, and just live with it. A glossy finish, while it, would look very nice in a kitchen, will also make it show up more. Your ceiling wouldnt bother me a bit either. It all depends upon what we are used to seeing, I guess.

A steamer will allow you to scrape it smooth.

I would want it smooth. lots of texture drives me batty. everything is smooth plaster walls where i am though. seems like it is pretty common in a lot of areas.

Actually, I have a peel and stick mural on a similar textured wall and it hasn't gone anywhere. The texture shows through, though, and it looks like vinyl. It would not be good here.