can't advise about the best paint but suggest you take one of the doors or drawers to a specialist paint shop and ask them (not basic hardware store)...you won't need much paint so the best quality to suit the job will be a worthwhile investment...should be a simple easy job but would imagine that brush painting will be safer and less messy for the walls but you could spray doors and drawers outside for a different finish but the paint specialist should be able to give best advice and would love to see the finished job and learn what product you chose

Where are you located? You have posted on the Australian Houzz site and I think Home Depot is in the US? If you scroll down to the Houzz logo at the bottom of the page you can choose another site.

Yes it was a mistake for you to choose a different floor for kitchen dining area. Replacement cost is a matter for you and your dad to weigh up. Firstly paint the kitchen shelves in the wall colour. Second replace the kitchen curtains with a blind. Handy hack to hold the chopping boards off window behind sink-install a slim towel rail below window. 4.the dining dresser. Paint it black or white. Can’t see table colour so may need to address that too. 5. Dining light swap for more modern shade. 6 add a dark strip of paper on the floor in the archway to separate the two floors. See if visually breaking the two helps. Then sit back and rethink the floor

I don't think a cream splashback will work if you decide on grey paint. I'd be painting a few doors to see what works - even if you paint the inside of the door and open it to get the effect.