Prior to our 07 remodel, we had cork floors in kitchen, DR, entry and adjoining powder room for 18 years. I loved this floor! We didn't treat it with much care and didn't receive care information when the cork floors were installed. So, they received a far amount of water when washed, no added finishes or refinishing. We also have a 22 pound dog who managed to scratch up a lot of woodwork around window areas. The floors held up well. If you'd look close (like eyes a foot away), you could see many scratches from a variety of sources. The excess water caused some seams to curve up a little (only noticeable with bare feet). I really wanted to put in new cork floors, but, got talked out of it by a KD who said colors/patterns weren't right for our new look. We put in porcelain tiles and standing on them hasn't yet been a problem. I probably don't stand on them for more than 90 minutes at a time. If I were to put in cork again, I'd use the floating kind and panels or planks. When replacing the old cork floor, everyone was predicting big problems if we had to deal with glued down panels. The old floor came up easy because no glue. Cork is much more forgiving on uneven subfloors than tile. It took about a days work to even out the subfloor for tile (involved replacing some plywood and a lot of sanding). One of my friends just put in a high end glued down cork tile and they now wish that they had gone with the floating option.

White Oak and Black Walnut both look great!

I was about to suggest the same. These are pine. They are not worth "saving" as a wood-look product. In fact the original attempt at stain is a bad one. Pine doesn't like taking stain. It really really doesn't. As Hallett & Co. suggests, invest in some HIGH QUALITY primer (Zinsser or better) and some decent sanding materials. Make sure you purchase primer that is SPECIFICALLY designed to go over oil based finishes/polyurethane. Paint the trim whatever colour you desire (I suggest white...it is much easier to coordinate than any other colour). Just by doing this ONE thing, you will have updated your home by 40 years. I kid you not.

First of all, I’ve been toying with the idea of adding a vestibule to the front of my home, so this project of yours is very intriguing to me! Moving on, since the current tiled entryway area is part of the room with the hardwood, replace that tile with the same wood. As for the vestibule flooring, will there be a door and/or significant change in elevation (i.e., a step) between the vestibule and the rest of the entry? If so, you could use tile in the vestibule. Would I recommend using the tile you’re thinking about using? Probably not, and for the very reason you’re concerned about—“that it would look like I tried to hard to get them to match.” But to know for sure, I’d really have to see the rest of your space and the plans for the vestibule, though my initial thought would be to use something completely different in shape, color and/or texture from the kitchen tile. For example, maybe use just those small hex tiles or perhaps use a charcoal/black tile. Btw, have you considered enlarging the opening to the kitchen?