How do I remove splatters of white paint from a hardwood floor wthout damaging the floor.?

It won't ruin the floor permanently though it would make it more work to refinish. However, it sounds like it's beyond refinishing anyway. Getting deep water or urine stains out can be all but impossible.

Isn't that what the flooring guys have already told you? If you prep it properly and use paint intended for a floor, a painted floor can last quite well and won't be any more slippery than most other floor surfaces.

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