Yes it's fine. The passive form makes it more polite, but you can also write it in the active form if you want to be more concise and deliberate. We (highly) appreciate your prompt action.

Yes, it's fine. In the United States, though, this has a formal tone, and sounds like it is trying to say something unpleasant in a polite way. This kind of language might be used by a business reminding somebody about an overdue payment. No matter how you word it, it's putting pressure on the recipient because you are expressing urgency; you are not only asking them to do something, but to do it promptly. If you are not trying to speed things up, if this is not the second time you are writing to the authority, you could say "thanks in advance for your attention to this matter." Finally, depending on the situation, in the United States, sometimes the best way to speed things up is to enclose a self-addressed stamped envelope for their reply. You wouldn't do this with a big company and a staff that sends out thousands of letters every day, it would just complicate things. But sometimes it helps, because you are showing them, in more than words, that you want a quick reply and are willing to do something yourself to help them give it to you.

It works and is, indeed, polite. If you're specifically asking a question you could say "I hope you'll find the time to answer my question(s)."

Yes, it's OK. You are telling them: do it fast.

It depends on your society and culture. It also depends if you have clearly outlined what you want the person to do before you added this note. Certainly in the UK it would not cause offence. But it might not add anything to the communication either.