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.
As others mentioned: key signatures are valid for all octaves. With one notable exception: scordatura.
Scordatura notation is sometimes used for bowed string instruments. In scordatura, the strings are tuned in an uncommon pattern. The notation, however, uses normal notes in a normal staff to indicate how a note is to be fingered rather than how it is supposed to sound (naturally this only works when the string assignment is unambiguous). In that particular case, the "natural" key on the different strings is different, and the key signature's accidentals are only valid at the pitch they are written.
But such scores are rather hard to come across.
Most scales are assumed to be octave-repeating, due to the way that we hear a similarity between notes that are an octave apart (the reason for this being that with many instruments, any note contains harmonic partials at the frequencies of all the overtones of a note an octave below).
This includes the diatonic scale, which is the scale that standard notation assumes (and that key signatures are most relevant to), and this is why key signatures are read as applying to all octaves. Hence why in your scenarios all 'A's will be flat unless otherwise indicated.
Incidentally, there do exist scales which are not octave repeating, e.g. http://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale.
(adapted from duplicate question Are tonal (sharp, flat and natural) key signatures octave specific?)
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.