First of all take a deep breath and try to relax. I've gone thru FeLV in my feral colony and household before and will try to offer the facts as I know them.
If a female cat with FeLV gives birth, the odds that all of the kittens in the litter will be positive is about 95%. On the flip side, if one is negative, the chances that all are negative are very high. Without having tested the mom, you have no idea if the positive kitten contracted the disease after it was born or from the mom. Kittens that haven't gone thru a full round of vaccinations are more susceptible for catching the disease so the positive one could have gotten it later. Some very young kittens will test negative to start and some vets will say that they need to be about 4 months old to do an effective test. I've read very mixed information on this topic.
Testing positive thru an Elisa test is not a conclusive test. The Elisa test is the one that nearly all vets start with and is more of a way to screen out exposure to the disease. However, many vets are not educated on the topic and many will make the statement that your sister's vet made: if one kitten has it, kill them all and the mom.
When a cat is exposed to the disease, their bodies try to fight it off but in the mean time, they will test positive on Elisa. It takes 30-60 days for the disease to turn into full blown FeLV if they can't fight it off. At that time, it goes fully into their blood stream and the only way to test conclusively is thru an IFA test. Unless a vet has a lab available to them at their office, they have to pull a vial of blood and send it out to a lab.
If they are able to fight it off, a subsequent Elisa test will most likely come out negative. That is why you usually run a subsequent Elisa test before you ask for an IFA test. The Elisa test is an inexpensive way to screen, and the IFA test requires a vial of blood and is much more expensive.
As far as infecting other cats: Vaccinations are not 100% effective and there is some speculation on how good they are. Statistics show that about 10% of vaccinated cats can contract the disease but only after either a violent encounter (fight or mating), or long exposure (such as sharing a house and grooming).
I do have a cat whose mom and all siblings were FeLV positive and was strong enough as a kitten to fight off the disease. Since you don't know about the mom, you really don't know the odds of your baby having it.
Take your baby back to the vet and have the test run again. If he is negative, don't worry about it. If he is positive, wait another month and have the IFA test run.
I'll send positive vibes that his test turns up negative.