My 3 year old daughter had the same issue. We used to take her to Little Gym for classes and she would spend the first 10 minutes clinging to her mother, who had to accompany her inside. But on her first day of Pre-K, the teacher kept her so engaged, she forgot about us. The secret is to avoid the "departing ritual". When they are first distracted, duck out of sight. If a teacher is engaging, the child will forget about the parents. Perhaps you've tried this and it didn't work. In any case best of luck to you.
I think that depends on the child. I keep nursery one Sunday a month at our church and it really depends on the child. Some kids you have to look in the eye and say good-bye. They cry for a few minutes then scamper off to play. Others need to have parents slink away out of their eyesight. Most, from my experience, do better when they watch the beloved parent leave. They cry, but do better after the tears dry up. But not all. Kids are frustratingly and delightfully individualistic.