I'm a strong believer in letting her make some choices and helping her follow her dreams of riding. I guess my questions are the following:
Horse riding is a lifelong hobby and for some folks an obsession.
1. Is 6 too young to start riding?
No a friend of mine put her son on a horse's back at 3.
2. What life lessons will learning to ride a horse give her?
So many that they can't all be listed here. It is not just the riding it is the handling and taking responsibility for another living creature.
3. How do I emphasize safety to her and the instructor without being too alarmist? You can't easily but nevertheless you must tell your child of the risks of hurting themselves. It is there and with horses it will never go away. Undeniably horse riding is classified as a dangerous sport. But life is like that. Make sure that your child always, repeats always wears a proper riding hat - with the kite marks to confirm it is up to the latest standard. Buy a padded body vest and make sure it is flexible and it fits. Get her to wear it when she is learning to jump.
If she goes out trail riding buy her a cell phone.
Choose your instructor carefully
I really want her to understand that riding horses is not easy, but that she can get a thrill by the challenge. Spending time at the barn has made me realize how special these animals are and I can understand why people fall in love with the sport.
Maybe you should get some horse riding lessons yourself - you are never too old to learn - except maybe me.
If I had children to bring up then I personally would encourage them to learn to ride, It is a unique sport. The child might one day fall off and break some bones here and there but most children heal quickly. There are plenty of other dangerous pursuits in this world and one has to learn to cope with them - and to avoid some of them.
If she ever loses the urge to ride, then don't push her on. But the younger they start, the better the rider in the longer term.
Buy her some horsey books and read them together.
Best of luck.