Correct. The ins and outs of who owns what can be complicated but as it's the Burbank Class C airspace it makes sense for BUR tower to be a controlling authority.1. Burbank tower would be the ATC authority for the Class C - correct?
Radar contact isn't required to enter the charlie, only "2 way radio communication". If the pilot called BUR tower and BUR tower replied to the pilot (without a statement to not enter and with the planes callsign) then you are good to enter.2. I'd say that technically the pilot busted the Charlie before this was completed because they entered the 48-SFC section of the cake before the radar contact was done.