If you play the Brilliant Administrator action card on a system that is subjugated, would you only get to use the first build icon and not the second? What about using it on a sabotaged system?

I don't remember the exact wording on the card but it allows you to place troops on the build queue using the systems resources. I am assuming correctly that a subjugated or a sabotaged system would reduce the effectiveness of those builds or does this card allow you to ignore those restrictions?

Without the actual wording, I can't be sure, but most if not all of those cards that allow you to use the build icons don't care about subjugation or sabotage. You get to use all of the build symbols on that system.

Kmanweiss is correct. Since its a mission, it follows different rules than if it were a build phase. Here is the text for reference:

Place this leader in any imperial system, then place units on the build queue using this systems's resource icons and number.