Do Russians actually use the diminutive forms of their names when introducing themselves?
Yes, in a proper situation (when it's not too formal and official, they are relatively young, it's a friendly meeting etc)
Kids on a playground was what this sentence suggested to me, I guess that would be an appropriate interpretation?
Kids on the playground for sure, what I meant to say is that it is not only for children.
Why is it Меня and not another form of "I"?
The literal translation for this sentence is "Me they call Vanya" "Меня" being the accusative form here.
Does it mean "I am called vanya" ?
Yes. Literally, it's "They call me Vanya."
"they call me Vanya " should be accepted as well
So, i would use this to introduce myself to somebody in a friendly manor or is ot still considered childish?
It's not childish. Many people usually go by diminutive forms in everyday life, myself included.
Isn't the они form of the verb to call звонят? Why is зовут used?
звать -> они́ зову́т
звони́ть -> они́ звоня́т