My Experience
I went through the waiver process for asthma to get into NROTC. I'm assuming you will go through OCS?
Yes, the cutoff age is 13. If you had no symptoms at or past age 13, then you are good to go. If symptoms have occured past 13, no matter how small, this is considered disqualifying by DODMERB, and almost certainly as well by NAMI (or NOMI? I forget which one is which).
You said you have some symptoms when you are around cats, which sounds like allergic asthma, not exercise-induced.
There is a small chance you can get a waiver for this. Like DevilDog said, PURSUE THE HELL OUT OF A WAIVER. They will tell you it's not waiverable; that's a bunch of BS. There is a process, it's just not certain that you will pass. You will have to get a Pulmonary Function Test with Methacholine Challenge. If you pass this, based on my experience, you will most likely get the waiver.
If you have any questions, just PM me I'll try to help you as much as I can.