What happened at the end of A Most Wanted Man?

First of all, you deciphered the situation quite correctly. Now why would the German and American governments betray Günther? It is true that they basically all share a common motivation, that is to counter terrorism and "make the world a safer place", but their means largely differ.

As Günther says himself, his group largely operates in a legal grey area and also with not much funding or backing from the government. This is why they have to rely on a network of informants and actual street work to get their information and achieve their goals. And this is also why Günther values and cares for his informants and tries to keep his promises, without his informants he can't do his job. He is somehow a spy of the old school, relying on defectors and double agents rather than brute force. And he has the larger picture in mind. He knows that Abdullah alone isn't worth much, let alone Issa who I think Günther was genuinely convinced to be totally innocent. He also says, that he believes in the soft methods of convincing people rather than just torturing them. So for him capturing those two is pretty useless. He'd rather want to convince Abdullah to actively work for him, a method that he has put to great success during the rest of the movie, all his achievements were reached because people directly worked for him (even if not always entirely out of conviction but also out of pressure).

The government on the other hand is in contrast depicted as largely bureaucratic. They want results and they want them fast. And they don't seem to make a big difference between capturing a small fish, like Issa who didn't even know anything contributory, and actually achieving a bigger result. This is not only demonstrated when they capture Abdullah, possibly trying to torture him for information, even though he probably doesn't know that much directly and won't be of much use. It is already demonstrated when Mohr wants to capture Issa as soon as he arrives and when Abdullah isn't even in consideration yet. All Mohr cares about is that Issa doesn't "plant a bomb" in his city and that everything goes strictly by law. He doesn't want a supposed terrorist run around freely, even if he is of greater use. The same goes for Sullivan and the Americans, they want Abdullah and his dangerous weapon business out of the way, disregarding if someone else will just take his place. They all don't have much trust in Günther's soft methods either. Those bureaucrats just want to get fast successes, no matter how small they are or if the invalidate further advances. So they just didn't even care that Günther's plan succeeded, they only wanted to sack Issa and Abdullah as a safe bet, the rest of the problems can be solved another day.

Now it is not entirely unambiguous to say if they intended to betray Günther all the time. But especially given Mohr's depiction and his aversion to Günther's methods throughout the whole movie, it's safe to say he never was a big fan of his plans. Now ambassador Sullivan was bit more nebulous, but I would also say she was largely just playing Günther all the time. Even if not with too much hard proof I would say yes, they never had much interest in Günther's plan at all. If the operation really took too long, then they could only be happy about that as it gives them an easy out excuse for their actions, but I doubt they ever had any genuine interest in its success.

1. yes
2. Gunther taking Muhammad into custody would have been just the beginning. Muhammad was a barracuda and now in the hands of the Germans/Americans he may end up in Guantanamo or somewhere where he will remain silent likely. Now Gunther's network is done. Muhammad funds terrorists, but he is just the tip of the iceberg. As PSH stated in the movie (paraphrasing) "You get rid of him, then you will just be knocking on doors in the dark"
3. i would say the latter

I just saw this movie. My take was that Robin Wright's character was tricking PSH's character all along. Notice how she conceded that the American's had really been at fault in Beirut. I think that was just to gain his trust.

They gave him 72 hours to get his man. They kept showing the clock. When 72 hours were up, they intervened. He was about ten minutes too late.