Ambassador Fox and his aide beam down to the planet even though the "Enterprise's" deflector screens are up - it was stated that transport could not take place under those conditions. However, Mr Scott had moved the Enterprise further out into space, in a wider orbit, where sonic disruption was impossible. The locals even acknowledged that they could not attack the ship. Therefore, it is reasonable that screens could be opened briefly for transport.
Ambassador Fox insists on making contact with Eminiar 7, even after the government there tells the Enterprise to leave. This is a clear violation of General Order One (commonly called the Prime Directive), and while the Federation diplomatic corps is a civilian agency and not part of Starfleet, the Federation civilian government presumably holds by the same philosophies as its military.
When Ambassador Fox beams down it is implied that he beamed down on his own authority without Scotty (in charge) knowing about it. Seems that a bridge officer would have immediately known if a transporter was being used without authorization.
The goof item below may give away important plot points.
When Spock rescues Ambassador Fox from a disintegration chamber, the door to the chamber is opened. As they back away to destroy it, the door is immediately closed. (Obviously stock footage of the first chamber destruction.)