Tony,
Read your additional posted comments..
what raises my hackles is...
"I have completed my side of the bargain"
and
"We will not confirm any class date until we receive payment. That is our policy, and it is non-negotiable"
Asking, who, when, where, what is only a fair question, if these cannot be answered then there is little interest in actually providing a service.
If dates are being filled and this outfit is in demand then these questions will be known. If not, then there has to be some degree of question as to the authenticity of what is being entered into.
If they cannot provide basic information then either move on, or take the gamble. you could of course always shove the funds into escrow.