#1
I fear that this is a very dumb question, but here goes:

I'm working on memorizing the notes on the fretboard--but I primarily play in Eb. Is it more advantageous for me to learn the notes as they would be in standard tuning or as they are in Eb? For clarity, when attempting memorization, should I call the Low E, 5th fret A, or G#?
#2
Both, I would start in standard, and then when you're in Eb, everything is just a step lower or whatever.
#3
Theory is about relationships, and those are the same regardless of key.
#4
Music theory isn't about patterns on a fretboard. An E note is an E note. An Eb note is an Eb note.

The patterns are definitely helpful, but irrespective of tuning they're quite useless if you don't know the key of the song, which will determine what scale you play, or ths notes on your fretboard which will determine where your shapes occur on the fretboard.
And no, Guitar Hero will not help. Even on expert. Really.
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#5
I would think Eb tuning as standard tuning and I wouldn't learn the note names separately for those two tunings. I would think my low Eb string as my low E string and if somebody says the song is in E, you would actually play like you played in F.

For example trumpet is this kind of instrument. When I play C on the trumpet, it's actually a Bb. If I have to read notation that is written "in C", I need to transpose a whole step up.

So yeah, I would just call the 5th string of the 6th string an A regardless of the tuning (unless it's something else than a standard tuning like drop D). IMO it would make memorizing the notes on the fretboard easier.
Quote by AlanHB
Just remember that there are no boring scales, just boring players.

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