>I am not really sure how large or small federal offices can be.

It's agency dependent. When I was Department of Labor, it was tiny cubicles for 11s and 12s.

With my current agency? You can tell how important someone is by the size and placement of their office. As a GS5 to GS7, my desk was in the hallway. When I changed job titles at GS7 I moved to an interior office. When I made GS11 I got a window office. My supervisor has an office 3x my offices size. My Director has a corner office 2x the size of my supervisors.

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>I am unsure what opportunity there is for advancement

This is very agency dependent. But a foot in the door allows you to change agencies eventually.

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>What kind of free trainings could exist, broadly speaking? Do you mean leadership training

Leadership training does exist, but also professional writing courses, online self-study, mental health, emotional health, general professional courses. Most last 2-3 days, some longer.

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>My absolute dream job (from my college days wargaming at GWU) would be to work in State at some small obscure desk

I'm unsure what exactly they would be looking for. That said, I mentioned networking and training opportunities. I've taken classes at the state department once or twice, and that would be a good topic to bring up during a break. There are also occasional opportunities for detachments, where you temporarily work somewhere else to broaden your experience. Usually it's withing the same agency, but not always.