The Galileo craft was burning up after Spock jettisoned the fuel, so why didn't it burn up when they first entered the planet's atmosphere as well?
Why didn't the annoying commander overrule Kirk when he turned around to save the Seven?
Answers to your questions:
1 - They never really answered that question as to why the shuttlecraft didn't burn up in the atmosphere, but then again, I believe they used what fuel the shuttle had to slow down the rate of descent. That's how space shuttles would enter Earth's atmosphere in real life. They would fire back-thrusters to slow the craft enough so that it wouldn't burn up from atmospheric friction. In fact, I think that's exactly why they had no fuel to get off the ground after the crash.
However, to claim they "crash-landed," when the shuttle is sitting perfectly upright on its runners, with no outside damage, doesn't look realistic in the least.
2 - I think that annoying politician who had been bossing Kirk around was not on the bridge when Kirk discovered where the shuttle was, and the Enterprise hadn't gone very far when the "flare" was sent up. I did like that the captain used a loophole to buy more time for the Galileo 7 to be found.
As for the first question, they never specified, but I would assume they had enough fuel to land. I can easily see that whatever pulled them off course didn't, in itself, affect their fuel supply.
As for the orders from Ferris, it could force them to leave. But once Spock set off the flare, indicating the shuttle's survival, he certainly couldn't prevent Kirk from acting. It would be difficult to explain to a review board that he prevented a rescue where the position of the survivor's was known and a transporter would work, in order to save an hour of transit time, regardless of the emergency on Makus III
What I found odd was that Kirk's standing orders to investigate an astronomical phenomena would over-ride a time-sensitive mission to deliver medical supplies. He could have come back and investigated after delivering the supplies to Makus III.
There is nothing in the dialogue to say that they did not have at least a somewhat controlled landing. In editing the production team evidently felt nothing else relative to the landing was necessary if it was filmed at all. My understanding of 1960's television is that a small fraction of what was shot makes it into the final edit.
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