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I'm sorry to hear about your cat. It sounds like she vomited up some roundworms, which are one of the most common types of intestinal parasites. Cats and dogs are born with them, and can be contagious amongst the animals who are "loose" amongst each other (i.e. share a litter box, or have microscopic traces of stool traced in on their feet, etc). Generally, this stays confined amongst dogs and cats. They even have their own species that affect them (dogs vs cats) but one rarer kind can affect both. The most common type of roundworms seen in cats is not seen in rabbits, and I personally wouldn't excessively worry about your rabbit and this problem unless he/she is out loose frequently amongst the cats and is showing any problems. Same with your parrots. I would just take the one who vomited the roundworms to the vet along with the sample, and treat the 2 cats for this problem. You also may want to double check your pup has finished shedding all ova and no longer needs to be dewormed by bringing his sample as well. Cleaning like you state is not necessary; once passed, they can no longer live long as they are not feeding on the intestinal lining. A general vacuuming would suffice.
Best of luck, but let me know if I can help further with the worm issue.
As long as all of you, including your grandson, use standard cleaning methods and wash your hands after handling the animals and before eating (or anytime for a 3 year old after playing or petting animals), there's no need to worry. The parasite of most concern to humans is hookworms, which are not what you are seeing, not commonly spread between animals and people, but do exist and therefore cleaning up before eating or before a child can put their hands in their mouth is always important to remember.
As stated above, there is no need to worry, the only way you could get this problem would be ingesting your cat's feces, which I don't see happening here. The passing worms are non-infectious and cannot harm you in any way.
The eye redness and lethargy sounds completely unrelated to any issue with intestinal parasites, and may be either allergy related or the start of an eye infection or upper respiratory issue.