I want to apply to be an unpaid intern in a government office and got arrest months ago (misrepresentation of age), fingerprinted and everything, but the charges were dropped. They took my social for the staffing application. What will show up? And how far would I even be checked out?

It would be available and view able by government agencies.
Depending on what agency and what position you will be doing is how deep they look.

Opinions about a congressional/senate office? Not police/courts or anything security related.

They may look, and want you to explain the charge, and what happened. Remember if something about you happens in their office, and the news papers get involved, all of a sudden, it becomes public to everyone in the world, and they have to explain.

Ok thank you

I doubt it will keep you from getting it, but it is very likely you will need to show them, you were not guilty , that you were charged by mistake ( which must be the case if the charges were dropped)

If they have already offered me the job before they took my social does that mean I'm in the clear or no?

No, job offers are always pending background review. But many intern jobs are appointed, ( political things) and they may not even do anything but a standard check of convictions.