I see two options. The first is to have your tile person straighten out the edge of the tile. Fabricate a reducer that's 1/4" thick i.e. the difference between your wood and tile), finish to match the floor and cut it to length. Undercut the door stops to the height of the tile so the reducer will slide under. Fill the gap between the wood and tile with something that's the thickness of the wood. Apply a little construction adhesive to the bottom of the reducer and install it, weighting it down until the adhesive dries. This can be done by closing the door and sliding wedges under it. The second option would be to use a matching baby threshold that would lip over the tile and hide the uneven edge. Fit it, fill the void, undercut stops and secure with adhesive. Trim the door bottom if needed. The bottom of the baby threshold might have to be shaved off so it sits level. If it's made for a 1/2" floor and your difference is only 1/4" you'd have to shave off 1/4". The first solution yields the lowest stepover height but is more work. The second is easy but has a bigger hump at the doorway.

You've done a very good job with the situation you've been given. The hardwoods look very close to identical. The only thing I could add to this - and it is ONLY an esthetics suggestion - would be to go 1/2 step darker on the transition strips. Like sewing a seam on a shirt/dress/pants, the ideal is a perfect colour match (thread to cloth). If the match can't be done, a good seamstress will go a snick darker. At a distance it is much harder to see a darker stitch then it is to see a lighter stitch. That is the only thing you could do at this point. And to be clear, the transitions are a pretty good match. If you do anything, the transitions going 1/2 step darker than they are right now would be all I would suggest. Other than that, think of those pieces of wood as the thing that saved you $15K. It's all about how you look at a situation. You can think, "Oh, geeze. I wish I didn't have to have those things there," or you can think "Thank GOODNESS for those transition strips! They saved me $15,000!" Still the same situation - just a different way of looking at it.

I prefer to stop the tile in the kitchen/nook. The hardwood can then flow from the formal dining area down the steps and into the living space. The ONLY thing that would cause any concerns would be the ability to find MATCHING stair treads for the hardwood you choose. Sometimes this is a BIG problem...if that is the case, then moving the tile all the way into the dining area would be the way around it. But you have to find this out BEFORE you choose/order your flooring. You do NOT want to make this decision on the fly...only to find out that you cannot find more of the tile in the same dye lot...that's WORSE than having miss-matched treads. Much, much worse. All of these decisions/questions MUST be answered WELL before you put pen to paper. A designer who is well connected in the building industry would be able to do all the leg work for you - without much effort. S/He would have already run into these issues DOZENS of times in their career and will know the ins and outs of working around these issues. And s/he will have ALL the needed professionals on speed dial. Which then becomes the NEXT dilemma which MUST BE dealt with before pen is put to paper. Whew! Lots of things to do before anything is decided. A well versed designer can be worth his/her weight in GOLD at this point.

The metal profile edge in your photo would be installed so that flange is under the tile, next the carpet tucks to it. This transition is located at the center under door when closed. That is where the layout plan begins as it's the most noticeable area. If this is a DIY project I can see it getting overlooked. If you paid for the tile install he / she can resolve the situation they created on their dime.