Wood floor is worn and damaged from water and wear any quick fixes?

You might try lightly sanding and putting a top coat on there to seal it. I would go with the same finish such as matte, satin or semi-gloss varathane that matches your existing finish. Feather it at the edge with a brush so it blends in, otherwise you can use a low nap roller.

To restore my wood cabinets, I cleaned them with KrudKutter from Amazon. It wil remove all dirt wax etc. Then I used furniture restorer. They make one for wood floors too. It made my cabinets look like New. You must CLEAN for an even look.

I agree with Sharon. But use a marine grade varnish in the same finish. Another option is to lay down some ceramic tiles by the doors about two feet out. You could even use some decorative tiles.

I wood putted the gaps- use a little paint similar colour to the u damaged timber. Once dry I rub in coconut oil- it came up perfectly. It doesn't leave a smell like other oils eg olive oil or cooking oil. Fantastic

If the sun hits that area, it will naturally bleach out the coloring. A rubber backed rug will also affect the wood coloring. Previous suggestions will work but might need added re-dos.

I would put a waterproof mat to catch wet and mud being tracked across the wood any more than it has. One both inside and out. See if you can find a min wax stain you can blend into the faded wood, then seal it with a hand rubbed polyurethane in a satin finish, easiest fix. Test a small area with the stain, sometimes you have to mix to get the right color. They make small cans of the stain of minwax. Good luck.

Assuming these are solid wood floors, not veneered? You can sand & re-stain & then coat with waterproof poly urethane for protection. If they are not solid wood, I would just put down a nice mat & cover the affected area.

Door Runner.