Can I paint fake wood cabinets?

Remove as much of the peeling fake wood grain paper. Just remove what is peeling. Lightly sand with a green kitchen scrubby to remove any bits. Prime with Kilz primer then paint your color. Seal with three coats of a water based polyurethane. You can use acrylic latex paint.

Yes, but first you need to either fix what us peeling or remove the film.

Oil based primer and paint would work out for you.

I’d clean them with a degreaser to clean well. Then use a primer and then paint. The primer will help the paid adhere.

Here's an article that might help you - https://www.bobvila.com/articles/painting-laminate-cabinets/

(Peel), Sand loose paper, Paint using chalk paint, Touchup. You’re through.