A veterinarian is using a new vaccine for cats. The vaccine has a 10% probability of causing side effects. In which distributions does the

I don't think the last answer in #1 should be checked.  We're given no information about a probability of vaccinating a cat.

One would think the vet would know how many cats they vaccinated exactly.  After all someone has to pay the bill!

The rest is correct.

In #2 the first two are indeed likely to be normally distributed, if for no other reason than that there are so many people entering the store the law of large numbers kicks in.

The rest of #2 you have correct.