I have some experience with this as well. Here goes...
When is the optimal time for a gymnast to begin ballet/dance?
I would say the sooner the better--it definitely can't hurt

. But if I were to say an age, I would say 7 or 8. At this age they are old enough to understand body awareness and follow directions a bit better. My dd started ballet classes at age 8 and then started taking 3 classes a week and pointe at the age of 11.
What is the best type of ballet/dance class for a gymnast?
Any beginner ballet or jazz class is good.
Where does your child attend ballet/dance classes?
My dd attends ballet class at a ballet school that teaches the Vaganova method.
[The Vaganova method of classical ballet encourages dancing with the entire body. Dancers trained by the Vaganova method will likely have high jumps and powerful turns, aided by the use of the arms. Many movements of the Vaganova technique require dancers to remain in the air for as long as possible to give them an illusion of floating through the air, which requires great flexibility and extension. The Vaganova technique encourages dancers to move their arms, legs and torso together in perfect harmony. A strong torso is a necessity for the Vaganova method, as the torso forms the foundation of all movements.]
I really like this method of ballet because it complements gymnasts very well since they are taught to use expression with their entire bodies. DD also attends jazz and lyrical classes at a competition school--they have a great performance show at the end of the year that all the kids love.
How differently is ballet/dance taught for a gymnast?
I am not sure how it is taught differently, but there are often more corrections that gymnasts are often told because of the habits of gymnastics, i.e. keep your ribcage down, shoulders down and try not to arch the back so much.
Why? Does taking ballet/dance improve a gymnasts skill?
Since my dd has been taking dance, her artistic expression in gymnastics has improved dramatically! Her core strength and balance has also greatly improved and she is a much more expressive performer.
*I wanted to add that one of the male leads at my dd's ballet studio was a gymnast up until 7th or 8th grade and then had to give it up to pursue ballet full time. His hard work and dedication to both paid off because his dancing is so athletic and dynamic. His talents also earned him a full paid scholarship to Juilliard for dance!!! He is starting his sophomore year.