First things first: find out the species of your wood floor. We don't have very many room shots of the floors...which makes figuring out the COLOUR they will become once sanded/finished in a CLEAR COAT finish. What you are seeing right now is not a 'stain'. It is oil based polyurethane that has ambered (turned orange) just by aging. I would say there is NO stain on the floors. They are a birch (maybe maple....TOO HARD to tell from these slices in the photos) with a finish over top...no stain.
The floor colour will drive the rest of the house. And yes...those floors in the kitchen are the very reason why we tell people NOT to put fake wood next to real wood. And the person who did it is probably colour blind. The clash of colours is enough for me to scratch my eyes out!
But I digress. Personally I would find out the floor species FIRST. Then I would get a hardwood flooring professional to come in and offer a quote for the purchase of 'more' of the same wood (in the same cut, width and grade) and the cost to lace in the two floors and refinish them.
That's going to be your 'big budget' ticket item. That will tell you what's left in the bank account. Painting out the trim is going to be a pain. The HEAVY graining of the red oak = doesn't look good when painted. I would assume you will need to remove/replace the wood trim unless otherwise convinced that something awesome could be done with them. I doubt it. Much of the trim looks like a patch-work-quilt that has been done a different times with different species of wood and different finish at different times with different skill levels.
The doors on the other hand are a different story. If they are solid core, then they are worth a bundle. I would keep them and try to find a shade of stain or paint that would look handsome with the FINAL look of my floors.
If you find out you have a VERY PALE wood underneath the orange oil based finish you might discover how pale they are when finished in a high-end two part water based finish (ahem...do NOT go with oil based finish or else you will simply end up with the same colour of floor = bright yellow/orange).
I've seen Birch and Maple as well as red and white oak look as pale as wheat or dry sand. It is a markedly different tone compared to what you have now.
Q
Hardwood over concrete advice
Q