No. Generally, these are for the dancers benefit and not for secret surveillance like you might see on TV.
If you are paranoid you can check the building for hidden rooms by wandering around or...
That depends on: The size of the studio
bigger studios, with more facilities, cost more.
The location of the studio
London will cost more than Cooper Pedy
When you want to hire it
Evenings...
To an extent, it depends on what you plan to record - if it's mainly electronic in origin (so microphones aren't being used) then you only need a computer with some recording software (Cubase,...