Considering these

Floors look great as they are. The gray "stains" look an awful lot like watered down gray paint, which would cheapen the look of your interior (because painted wood floors look cheap and bad). If you want gray, paint the walls - natural wood and warm gray is a nice combo.

In my honest opinion nothing but nothing in any gray. Just do not like it, why hide wood like that?

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Oh wow, so many negative opinions. I wanted to go with the Sunbleached stain for living/family/hallway and some cool darker brown for bedrooms.
The floor is made of red oak so all brown or natural stains are making the floor redish (hot) and that is mainly why I want to get rid of the original color.

Maybe someone has a tested stain that is natural and cool, and would look good on red oak?

I also like something like in the pic below but this is impossible with red oak.

First use the same stain on all the floors and if you like dark use a dark brown stain I lobve gray but not on those floors for sure.This is Jacobean stain on red oak I think a very nice darker floor with gray walls.

To avoid pink undertone on red oak you have to go dark brown. Some mix special walnut and jacobean.

These are both on red oak. Jacobean and Special Walnut. If you mix the two stains, you end up with something in between. So darker than medium brown but not dark, dark. And no red.

EAST HAMPTON: NEW INSTALL & FLOOR RESTORATION - JACOBEAN STAIN/TRAFFIC HD POLY · More Info

Aquebogue new construction - 3 1/4" Red Oak installed / stained Special Walnut · More Info

What about something lighter?

I have a light haired shedding pomeranian puppy so not sure if dark floor in living space is a good idea.

Red oak has strong grain and going light (natural) is not going to deal with the pink. Same with whitewashing. You can have them bleached but will still have the strong grain. And you're right that dark floors are a PITA with light pets and just show any dirt much more. Part of your problem is the existing sheen. Matte is the way to go. with water poly.

My red oak floors (which were quite 'orange' before) refinished in jacobean. We love them, but yes they show more dust, etc.

I agree with everyone's advice here and I am someone who usually likes gray. I think gray hardwood only looks good if it looks like the wood naturally aged. Wood is a thing found in nature, so embrace its natural warmth.

I really like suzyq53's inspiration photos for your floors, but you could probably do a custom mix that is a medium cool brown if you think those are too warm. I just think your current stain color samples are too grayed out and because your floors are red oak, the lighter you go, the more pink it will look. Also, you should choose a stain color you LOVE and not worry about the dog hair so much in influencing your floor color. They are floors, meant for us to walk on, and are going to get dirty regardless of your shedding pup's hair color. Instead of trying to hide dirt and hair on the floor, just come up with a good cleaning schedule -- maybe get a robot vacuum that can clean the floors everyday while you're out.