Would this work for an older dog that is having a hard time getting up from a tile floor and needs some traction?

IMO would be better than tile I would think.

This is more like a moving blanket.

Side of blanket that touches floor has some grip to it to help from slipping. Top is softer “like a moving blanket”... soft not wooly

Our pet soiled these pretty badly. Can they be washed in hot water with oxy clean?

Do not use very hot water… wash in washing machine medium setting with warm water. Yes on the oxi clean detergents but do not bleach. Tumble dry on medium setting (warm)

So I have two puppies that have shredded any blanket, towel, mat, dog bed. Will this be the same?

I had four pups use this during the day along with their toys have not had any issues with them chewy on it. Your pups may just need more chew toys with them and them again may just be chewers while they are teething

Can I use this as a blanket over our bed quilt to protect quilt when bunny is on bed?

Yes. If they potty on it, it doesn't soak through so it won't mess up your blanket.

Will this help the pen to stop sliding around on a tile floor?

It stopped my pen from sliding around on the floor. It also washes and dries very nicely