I would think you need to rip up all the damaged wood flooring and see if you have

structural damage underneath. Then I would replace with wood. That way you will know right away if/when you have a water issue. Putting down tile, may not show water damage, unless it floods over the tile. Perhaps a more knowledgeable persons could answer your question.

Also, do you have insurance for this kind of thing?

Yes, the previous owners did have pets, but it just looks suspicious since it's all along the front of the house! I'm not in love with the marble in the entryway or the parquet so was hoping to replace all of it since there's different wood flooring in every room in the house and I like some of the hardwood floors in the bedrooms more (not to mention it would be more costly to replace the floor in the bedrooms so everything is consistent). Will engineered wood stand up okay with an entryway with super heavy use or should I go with a natural stone / tile and bring it into the living and dining room space so it looks less choppy?

I do like to have something other than wood in a foyer, especially if it is a heavy traffic area, you're in an area with lots of inclement weather, and room layout permits it. Your entry makes it easy to do something other than wood since it's separate and defined. That said, I have wood in my foyer and it's not even very resistant to heavy traffic but few people enter there and we do have a covered entry porch.

Pets will often find a discrete place to do their business. If the prior owner had furniture situated where a pet could go behind it and relieve themselves that might explain the concentration in that area. Also they often go on area rugs so you might be able to see the outline of the rug.

What you put in the LR and DR is a personal choice, so hard to weigh in since it's subjective. I suppose I'd determine whether I like something enough to have it in the entry and the other areas. Having one floor would make the area seem more spacious while having different floors could define particular areas and allow for different design treatments.

There is no fixing water damaged wood.

Whatever you use wood or ceramic in the entry area, you will need some kind of rug

to wipe dirty shoes on.

"There is no fixing water damaged wood."

If a large portion of the floor is good and replacement wood is available, bad portions can be replaced and the whole thing can be refinished. Some might call that fixing water damaged wood.