Please do not let the cat become and outdoor cat. First, because it has been an indoor cat and will be at additional risk for all the normal things that kill outdoor cats (cars, wild animals, disease, mean people, etc.) and second, because outdoor and feral cats are one of the most harmful animals out there for native wildlife.
Many (if not most) instances of inappropriate urination in cats are due to medical issues, some of which can be life threatening (urinary blockages, diabetes, and renal failure are examples) and pretty much all of which are treatable. Ask your parents if either of them has ever had a urinary tract infection or kidney stones. They are very painful ailments and are painful for cats too. Except that cats cannot reason out the cause of the pain, so they associate it with the litterbox and start avoiding the box to get away from the pain.
Please keep pushing for a vet visit with a urinalysis (they are not very expensive), if your cat is sick it is acting out of pain, not to spite you.
For puking, if you have reason to believe it is hairball related (hair in the GI tract can make them puke, even on days they don't puke up a hairball) I'd suggest either a standard hairball paste like Petromalt or
these which I've personally had very good luck with. Getting the cat on a better food can also reduce hairballs tremendously. My little calico used to cough up a hairball EVERY day until I put her on Innova EVO. Now it's less than 2 a month, even though I do not brush her. If I use the hairball treats it's even rarer.
In addition to the stuff suggested above for fleas you can get a pill called Capstar from the vet which will kill all the fleas currently on her in 24 hours. After that you maintain the flea free state with one of the monthly treatments. Under no circumstances should you use the once a month flea stuff from the grocery store (Hartz and the like). Not only do they NOT work, but they can kill your cat or make it very sick.