I'm not very familiar with how US politics works internally, but how would it play out in practice? My experience with my (admittedly flawed) government is that the head of such agency would be dismissed from his position, and a new - more amicable - one appointed not long after. Are the US different?

It usually plays out like this.

The Congress person from Alabama where the fuel tank is built refuses to ok spending for NASA's other projects (or some other desirable project that needs to get done) unless the design requires things be built in their district. Since the tanks were previously built there it becomes the easiest way to satisfy their black mail. This may not be explicitly stated other than in meetings with the speaker of the house but it is understood none the less. This is not Congress people directly profiting from this decision but they have to run for office every 2 years and need to have consistent pork returns to keep their constituents happy.

There is nothing illegal here in fact the system is pretty much designed to work this way to insure that Federal money is distributed among the states.

Congress dictates what executive branch agencies do, not the other way around.

Notice, however that it's not about _what_ to do, but _how_ to do it. And when it comes to to the how of human space flight, NASA has the experts, not Congress. Politicians rely on Engineers and Researchers to make informed decisions, it is therefore only a natural expectation on behalf of Congress that unreasonable, meaningless, or dangerous requests should be denied. If that doesn't happen the Administrator screwed up, it is that simple.

I think what you mean to say is congress approves the appointment of officials nominated by the executive branch and approves the budget for agencies. Agencies have no control over congress. The executive branch can dictate what an agency can do, within the boundaries of a law originally passed by the congress (though almost never a recent congress).