Which again leads us full circle to "Why did he install it in the first place?" The theoretical answer is that the Enterprise cannot be operated purely from the bridge unless the rest of the ship is placed under automatic control. If the automation fails, the ship cannot be controlled.The problem isn't why Scotty hooked up the automation center but whether it could be manually overridden when broken.
You cite "doomsday machine" that the ship CAN be controlled from a single point, which would render the automation center on the Enterprise entirely useless. There's a contradiction there that needs resolving. Either Constellation possesses a capability that Enterprise lacks (and therefore the latter cannot do the same without being configured for it ahead of time) or -- more likely -- the automation center is standard equipment and Decker activated it just prior to beaming down his crew.
It is, actually, because the STARBOARD side of the saucer is lit up here while the PORT side is not; that side of the saucer is shaded from the light effects by the bridge dome and the upper levels of the saucer itself. This means the impact point is somewhere on the starboard side of the saucer.The whole back half of the saucer is lit up because of the glow from the expanding gas that wraps around the back of the saucer. That isn't indicative of where the hit is.
For the duration of the explosion and its after effects, yes. Then, as soon as the blast and the secondary effects dissipate, the shadow returns to its original darkness, cast by the impulse deck by the external light source to the ship.And that is where the problem comes in for an impact directly between the impulse engine and bridge. The front of the impulse deck would be lit continually...
Which is exactly what we see.
But you cannot see the impact point, which is obscured behind the bridge. If the torpedo was moving down boresight (the camera is in line with the torpedo's flight path) it would be still be impossible, since the torpedo would have to either physically pass through the bridge dome or otherwise "dip" below the visible structure of the dome/saucer to hit that point. Indeed, it could only hit the point you specified if it was traveling (in the camera frame) from roughly top left towards bottom right.Sure it can. You can see the port side of the impulse deck from that camera angle...
From the camera's point of view, the impact point is slightly behind and to the left of the bridge module; IOW, outside of the red lines on the diagram below.

Note that the path of the torpedo is from top right to bottom left. At that trajectory it could not have crossed over to the port side of the saucer section before hitting the ship.
Actually, that's something they spend most of their time practicing in the simulator.Operating a transporter to select a beam down point to beam a survey team down and retrieving them would be one of the things a cadet crew would need to practice on a ship.
Training ships are used to teach cadets how to live and work aboard an actual ship on a day-to-day basis; transporter operations and away mission scenarios are trained on a situational basis and such training is usually conducted in a simulator or (in the field) in the form of drills and exercises. "Learn how to use the fire control system" is simply not what training ships are used for; "Get used to using the fire control system on an actual ship where you don't have your academy bunk, the bar across the street, the library, your girlfriend or your momma's cooking to keep you comfortable" is.
That raises the question of whether or not the Enterprise was actually under Kirk's command prior to the Praxis explosion. If he was, then "under your command" would seem a little redundant, wouldn't it?The status of the bridge crew of the Enterprise need not indicate the status of the Enterprise herself. If the Enterprise was not considered able to protect the Klingon delegation 900 light years to Earth against Klingons or Earth ships trying to disrupt the talks then she would not have been sent. They could have easily had Kirk take over the Excelsior or another ship if the Enterprise was not up to the task.C in C: Well, there are Klingons who feel the same way about the peace treaty as yourself and Admiral Cartwright. But they'll think twice about attacking the Enterprise under your command.