The lowest jumps I have seen are 12 inches, but that is hardly seen. Most places will start at 18 inches but these classes are normally only seen in the Hunter divisions rather than they jumper divisions. Jumpers normally start at 2 foot and go up. I have also seen poles placed on the ground that riders go over in classes.
As far as age goes, there normally isn't an age that you cannot do low jump classes. There are leadline classes that I've seen have the ground poles for tiny riders. I think that as long as you fit the classes guidelines than your age doesn't matter. In the hunter ring there are three age groups, but I'm not sure if the age group has anything to do with the jump heights. I know there is short stirrup for riders 12 and under, long stirrup for riders 13 to 17, and rusty stirrup for anyone over 18. But again, I don't know if your age has to do with how high you will be jumping.
Hope this helps. Also most shows have a class rules and descriptions posted with their class lists, although I'm not sure if you are looking at showing or if you just wanted to know.