I have this in my house, done by P.O.'s. It is all in bed, bath and hall areas. In one bath the inside is white trim with it stopping at the big board trim piece that the door shuts against--that piece is stained like the hallway. The door is white on the bath side and stained on the hall side. the door edge is painted on the lock side and stained on the hingeside-hallside. I also have doors in the bedrooms that have been left stained on both sides with the trim white and I actually like the look. HTH

Some might scoff at real stuff next to fake, but I have a few areas of real next to fake in my home which look fine (stucco next to hardie board, real marble next to fake ceramic marble tiles, and wood tile next to real wood to name a few). My wood next to wood tile is running the opposite way and I think it looks fine and was a good tradeoff considering the functional improvement tile provides in combined kitchen/mudroom area and the difficulty of matching 100 year old douglas fir, however 1) The opposite direction run makes sense for the orientation of my kitchen, that is the tile runs with the long dimension of the kitchen 2) Its at natural room transitions, not an abrupt change in an open room 3) The two woods are clearly different from one another so not trying to "match" per se and missing but they are in the same tone family, that is (to my eye at least) both are a neutral brown without any significant grey/red/etc. 4) I carefully picked a tile with some texture to look similar to the 100+ year old original floors. In your case, I'm not sure it works so well. Seems like you have grey "wood" joining with the actual reddish wood in what seems to be an open(ish) area. Also one is sort of a distressed rustic sort of look whereas the other is a shiny smooth (I assume prefinished) look. Seems like you already bought your tile, not sure if you have the opportunity to still change your mind. If you are 100% committed to the floor, I'd do them opposite directions.

Shim up the nosing to the tile height. Add a scotia to cover the gap on the underside. The tile installer could have taken care of it for you.

I think it’s permanent. When we measured with laser it was completely straight line but after laying tiles it starts flash with floors but goes about 1/4 inch gap to wards the end. I think by the time we get to the end it will be more than 1/4 inch. I was thinking to take off more about 4 inches of wood and lay a piece of sand wood opposite direction to make a transition point but that will probably only work if time end up flash with floor height that I will see only after wood is sanded.