When Ambassador Fox's aide is struck by a disruptor beam (how, since he was behind everyone else?), The Ambassador just checks on him and goes on - no report to anyone that he was hit. There is no indication of the aide's state - live or dead - he just seems to disappear for the remainder of the episode.
When Kirk orders the 'General Order 24', as the guards are trying to restrain him, his hand bumps the central box on the desk with the two mini-view-screens. The bump moves it - indicating the equipment on the table is merely resting on it, and not built-in, as would be expected.
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 Enterprise initially enters orbit because the ambassador disregards a 3-digit code intercepted outside communications range. But the planets have the technology for a sub-space data link between computers several planets apart, so they could certainly have explained the situation in more detail before the Enterprise entered the war zone.
Scotty moves the Enterprise to a higher orbit, out of range of the planet's weapons. But the Enterprise has already been hit by a simulated weapon from another planet. So the weapons must be able to hit several planets away - a ship in any orbit would be in-range of the planet it's orbiting.
Regardless of their local agreements and war rules, any war simulation would necessarily involve all affected parties being aware of the simulated launches, and the simulation would have to calculate each party's capabilities. And since the Enterprise had the technology to detect, evade, or destroy the incoming weapon, the simulation would have shown it to be undamaged.
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.)