Do you guys think it's a good idea just to put her litter box in the tub
I think TopCat3 is right, try putting the litterbox IN the bathtub. If you don't THINK she has a UTI, it's best to get her to the vet to rule it out.

The litterbox should be in a quiet place, free from alot of "traffic".

BTW, cats don't necessarily need to be outdoors to get fleas. Actually, you can bring fleas in on your shoes and clothing.