Birch

The plywood panels are beyond any doubt birch. The frames seem most like maple to my eye.
Casey

Agree the panels are birch. And the rails and stiles do look like maple.

Repairing the counter trim is possible. It will be easier if the trim can be removed without breaking it. I see nail holes and hopefully there is caulk and not glue in the space between wood and counter.

Either way, all the old finish has to be removed. Sanding that would be messy in place. But, so would stripping. Again, better done by removing and working it in a garage or shop.

Get all the old finish off and try the stain on the back to get the color right(sand the test area). Then stain and let dry. Sand(by hand) with 220 grit very lightly(knock down the nubs).

The reason the old finish looks so bad is that it is probably polyurethane. That chips when hit. Fixing poly is very difficult.

I'd use a oil based varnish instead. That can be spot sanded and retouched if damaged.

Staining the cabinets is really not worth the time/effort. All the old finish has to be removed. Sanding there is not really an option, except as the final prestain step. And, since the two woods are different, the same stain will produce two different colors, both of which will not match the counter trim.