There's no such thing as "tile and grout" per se. There are worlds of different tile *and* grout now so that you have to make sure you're comparing apples to apples. Grey grout is a wonderful thing. In the 90s, I spent way too much time on my hands and knees cleaning white grout between the cheap white ceramic tiles in the bathrooms of our last house. HATED it. We also had less than 10-year-old vinyl in the kitchen (I didn't choose it--the PO did) that had yellowed. Fast forward to our new house: we have put gold/beige throughbody (same color all the way through -- great in case of chipping, which hasn't happened) porcelain tile with either Laticrete natural grey or TEC standard grey grout in the baths and laundry/mud room (it gets muddy in Oregon, esp. with an outdoorsy family) and it's practically bulletproof. If you want even more bulletproof, get your installer to use epoxy grout (not in changes of plane, though! caulk only there!) and dirt won't have the slightest chance of sticking. The only thing that would move me in the direction of a 'resilient floor' would be the softness underfoot. If that is important, and it's a nice feature as we age and our knees hurt more, I'd go with that. Vinyl has also improved its technology such that the higher end ones are much more durable in terms of color and puncture. You could underfloor-heat wood or tile, but not vinyl, as I understand the current technology. So I'm throwing some other factors in there, and trying to make sure you are judging on specifics.

the ozone machine may work, but i have my doubts. as pet stains that are persistent usually are the ones where the damage has gone deep into the floor or even the subfloor. Sometimes the sanding and refinishing will take care of it. Sometimes it doesnt. My suggestion is check the prices of the air purifier and if its not going to break the bank try it, it cant hurt. But if my suspicion is correct, then the only way to remedy it would be to remove the effect floor boards and install new. a good pro will have no issue doing the patch work, the sanding and blending of the stain to match requires a really good pro. but worst case you resand that room, and by the sounds of it, its not that big, so not a Huge ordeal.

A two story home should take 3-5 days (depending). The floors in the home look to be red oak. They will have a natural tendency to have peach/salmon tones when 'raw'. The look you are after is 'raw'. Do not use the word 'natural'. It is confusing because in the wood world 'natural' is the colour of wood with oil based poly on it = orange colouring. By using the words "raw look" you will make it clear that you want a wood floor that looks sanded down with nothing on it. Now for the finishes: DuraSeal = low-grade (ie cheap) finish that is rated as professional...but just barely. Bona is a true professional grade brand. It ranges in price (low price = Mega = 1 part cheaper product = turns yellow/orange). The highest price finish = Bona Traffic HD. The HD is the TOUGHEST finish on the market. It comes in 'commercial extra matte'. It gives the look of raw. The oil based products = HUGE odour issues. Which means you will need more than a few days before you move in. The odour is enough to cause migraines and nose bleeds. And it takes as much as 30 days to cure....which can cause BIG delays when moving in. Water based products (the real water based...not waterborne) are clear and have very low odour. The big guns, like Bona Traffic HD or Loba 2K Supra AT have odours that dissipate inside of a day. They take 5-7 days to cure...which means you can move in 7 days after the floors have been finished. You are looking for a raw look finish. You are looking for professionals who KNOW how to work with the expensive 2 part products like Bona HD Traffic. You are looking for refinishers who are willing to spend money on the high end finishes (which means you are willing to spend money on the high end finishes). And it traditionally takes 1 day to sand down everything, tack cloth it and then let it dry. It takes 2 days to properly coat the floors in a total of 3 coats (two coats per day = maximum amount). I think you can kick the 1 day dude to the curb. A single day is ONLY possible if he gives 2 coats = going cheap. Do you LIKE the heavy orange tones??? If you do NOT like ORANGE then you want to stay away from oil based finishes. Remember: 'O' is for "Orange" and "Oil based".

The wood-look tile has a specific "brindle" (stripe) look to it. That is rare in wood flooring. You can find it in Ash and Hickory and some lower grade Maple. The stripe is something that is there or it isn't. Your stairs do NOT have it. The pattern/look will NEVER come. Ever. Now to the COLOUR. The gray is going to be SERIOUSLY difficult to do on red oak stairs. Super-duper difficult. As in, "Tear your hair out...wailing...." type of difficult. Which leaves the PALE skin/flesh tones. That's MUCH easier for red oak. When red oak is sanded down (ahem...palm sander on stairs = not a lot of fun = plenty of $$$ to a pro) it turns this pale flesh tone. Or very close to it. This colour can be preserved by using a high-end water based finish like Bona Traffic HD (satin would go well with your tile gloss level). What will that look like with a CHUNK of flesh toned stairs in the MIDDLE of a zebra striped floor? It will look ... how to say it, how to say it ... gawd awful. The fix for a skin toned stair in the middle of a zebra forest? Stair runner in a gray/cream colouring. Sigh. So...the REAL fix would be: Find a stair runner that works with your tile. Have the carpeting company come in and do the measurements and suggest the WIDTH of the runner on the treads. Find out how much of the tread will be visible on either side of the runner. Now you bring in a flooring person to strip the OUTER EDGES of the stair and finish them (yep...leaving the old finish in place....it will save ++$$$). Have the carpet runner installed and off you go. But to be honest with you, I would recommend another tile. The sample you are holding has 3 repeats in it that I can see. That means there will be HORDES of repeats throughout your floor. And the striping will be the Elephant in the room. You will see nothing else but the stripe. It will be busy, busy, busy. And then, it will be busy. Right after that...busy. You might want to ask yourself what it is about the tile COLOUR or PATTERN that attracts you. If you can narrow it down, you might be able to find a solution that isn't quite so....busy. And that option might be easier for your stairs to match.