I'm surprised that the shelter didn't bathe her and clean her eyes, get rid of fleas, etc. Do you know how old she is? If she is real little, you will probably only want to use baby shampoo on her. If she is older, you can use flea shampoo. As with dogs, you need to treat the pet and your home. A trip to the vet could take care of her rabies vac (if old enough) and you could get some drops for her eyes. Although I would not rush to treat her eyes unless the discharge is green, it may just run it's course. And heartworm... hmmm that's a good question.It seems to me that cats should be on heartworm prevention but I'm not positive.