1.As indicated, I placed the flow inside the music frame, but it extends outside the frame.
Increasing the bottom value positions it inside, but adjusting the top value moves it outside.
How should I set up the frame?
2. Can I set staff labels to “full” on the first system overall, but set specific flows or instruments to “none”? Like in the score above, where there are no instrument names but the spacing appears together. Even though the full name field is empty, it still gets indented.
I think you probably misunderstood the terminology and usage of flow and music frame. Flows are created in Setup mode. Music frames are created and edited in Engrave mode.
You can set a music frame to display a flow or multiple flows, through its filters.
I see here two music frames (the blue outlines). One hast the music frame chain LAP (the L tells that you created this directly into the layout and not in a page template). The other music frame show only its edges in your screenshot. So not sure what the situation is.
You probably need to extend the height of the little music frame, to accommodate that system, or reduce the Music Frame Top and Bottom margins in Layout Options>Page Setup>Music Frame Margins
Not sure what “inside” and “outside” means, without looking at the Dorico project itself.
You can create Breaks (system or Frame breaks), and in the Properties panel, you can set different Staff Labels visibility options: