I'm learning a song in the key of D flat. I know that any middle A within the song is flat. However, is a high A (above the staff) still flat?
Yes. The key signature of Db has a Bb, Eb, Ab, Db, and a Gb. Those notes are flat unless otherwise noted no matter the octave.
For any key signature on any staff, you will only ever see the accidentals written once in a typical pattern. The octave the accidentals are in are entirely based on the clef used, but apply to all octaves.
You can think of the key signature as a general guide to what notes belong to the key you are playing in. For example, in the key of F the key signature has a Bb in it. So it is telling you every B is flat and all other notes are natural in the key of F.
If a flat or sharp occurs in a key signature, it means ALL notes with that letter name are affected. Not just the ones that are on the same line or space as the one in the key signature.
It helps to remember that key signatures were originally invented to save time, paper, and ink back when all music was copied by hand.