It is pretty easy to get a decent match on older wood - if you are willing to pay! There are numerous companies that specialize in reclaimed lumber. They would be able to match you up with some planks from the same time period and mill them to match yours. We just did that with heart pine in a 1912 house. It is not cheap though. Lumber, milling, installation/weaving, sanding and finishing came in over $25/sq ft. Fortunately it was just a few small patches. If you are just trying to make a new board look "old" - good luck. Modern lumber just isn't going to have the same look as the old growth trees they were logging 100 years ago. If you can swap out some boards from a closet for "free" that is a better option. As for the costs, ripping up 2 sections and repairing them obviously will cost more than just 1 section. For the urine damaged areas - whose idea was it to just sand? If you asked him to do it to save some money on repairs and now don't like the look, expect to pay to have it fixed. If you left it up to the contractor as to which boards to replace and which to salvage and he just did a poor job, then he should stand by his work and fix it.

The weave in looks OK, but there is no excuse for that result, other than not being able to source matching material due to cost or time constraints. I wouldn't want to live with that also, and I would never expect one of my customers to think that's OK. But it is, as you say, a conversation piece. Even if you source very similar material in terms of species, sawing, age and location, a flooring expert/artisan is still going to have to tweak the new material with stains and dyes to make such a repair less noticeable.

Wood needs room to expand and contract due to seasonal changes in humidity and temperatures. While it is certainly possible to do what you are proposing, filling the gap between the tile and wood with caulk is not a good idea. There should be at least a 1/4" to 3/8" gap between the two. That means a transition strip will be needed to cover the gap, one that is connected to the sub floor or the tile thus allowing the wood floor to move. If that transition strip were made of metal---aluminum or steel, it could be made thin enough to not cause a tripping hazard. If it were made of wood, the thichness will need to be at least a total of 3/4" above the floor surface---tapering it will limit the actual trip hazard, but it will still stick up that 3/4". That 3/4" is basically the same as a threshhold at the bottom of an entry/exit door or often the difference between a carpet and tile floor between a hallway and bedroom---with the transition strip in a doorway. Those locations do not cause the same propensity for a trip since there is a definate change of location---going from one space through a door to another. People have learned to expect a change in the floor height in those conditions. If the change in the spaces is enough, a 3/4" transition strip between the tile and wood could work fine---it just depends on the situation.

We don't have any plans to put a rug in where the transition area is. The area rug we had in there before was on the old wood side. There really won't be anything covering that space. On the step up area (which you don't really see in the picture), we also had some new wood laced in with the old (from the removal of a pony wall). The wood flooring extends from the end of the tile in the kitchen through the rest of the house. There are 2 boards width plus the step down edge and they actually blend a bit better. I assume part of the problem is the large space. @Jan Moyer - this is a great room, and other than the tile kitchen, the rest of the house is wood flooring (sans bathrooms). The step does suck, but we're living with it. The potential to change the stain on the great room floor would accentuate that there is a level change. There are, what I believe, are sander marks anyway, so re-sanding probably needs to happen. Going with some test stain patches is the best way forward I think. At least we'll have a better idea of what we'll have to live with. I'm just hoping we can get some warmth in the floor color without the more dramatic transition change (it is a bit more than what the picture shows but the picture does demonstrate the change).