What started ______?
a) you scuba-diving
(b) you to scuba-dive
The answer is (b).
Q1. Why can't (a) be an answer?
Who says (b) is the correct answer? Native speakers would say (a) and NOT (b)
It might be a good idea not calling your child shy in front of other people.
-> It might be a good idea not to call your child shy in front of other people.
Q2. Why should calling be changed to to call ?
It is the form of the sentence you have used.
Say this incident has just happened, and as the grandfather, I say to my daughter:
"Calling your child shy like that in front of other people is not good for the child's confidence."
I am referring to a specific situation that has just happened right then and there.
In your sentence, let's add a few words:
"As a general rule of child rearing, it might be a good idea not to call your child shy in front of other people in future."
Here, I am speaking generally, and suggesting that when situations do occur where the child is shy, at least, don't comment on it to others in front of the child in future.