Your appliances and flooring are both warm and will not do well with a cool gray. Look at warm gray or greige paint samples with those items before painting the cabinets. Be sure to paint the brown trim around the ceiling also.

What about a brighter paper with a bit of brown?

Cole and Son Prism

Does anyone have any experience with painting old appliances with enamel paint?

I like palimpsest comments. Paint the walls and the ceiling trim first and find a nice patterned fabric to replace the café curtains. You might find that you are happy with the update. The cabinets are a nice wood, not the hated orange oak, so that is a big plus fro you.

Fabric with white, brown and a teal/blue would be pretty in your kitchen.

If you mean the dishwasher, there are screws on the sides holding the front panel on. The panel is usually black on one side and white or cream on the other. Pre-remodel, I used the Rustoleum spray paint to paint my old one silver. If you want to give it a try on the non-black side, look for a paint that matches the brown.

I noticed the saved photo in your idea book with the dark grey cabinets. Is that your inspiration? Although beautiful I don’t feel you’ll be able to achieve this look successfully. You have to work with the elements that are staying, flooring, wall tile, appliances. They suggest a retro style kitchen. You can still refresh with more of a modern flair, but still keeping the retro feel. I like the wood on your cabinets too, but if you do want to paint them, maybe darker Greige lowers and white uppers would be an option. See photo below. Not a big picture but I like how the tile gives the retro vibe and cabinets look modern.
If it were my space rather than painting I would put my $$ into some nice retro tile, new hardware, lighting, paint and window treatment.