As the title suggests. When playing the Dorian flat 2 mode in Ab, the second note would be called a Bbb right?
Yes, Ab dorian b2 or phrygian ♮6 has notes: Ab Bbb Cb Db Eb F Gb.
Consider that this scale is a second mode of Gb melodic minor. Name Gbm is actually rarely used due to large number of accidentals, F#m is more frequent. If you switched to sharps, you would get G# dorian b2: G# A B C# D# E# F#, so a bit less accidentals. But if you have a good reason to call it Ab scale, then yes, that's correct.
B♭♭ is enharmonically equivalent to
A natural which is one half step above
Why, the crazy spelling with a double flat?
Try thinking of it this way: if you numbered the degrees of a scale, each degree would get a separate number. For a seven step scale like the one in questions you would simply number it
1 2 3 4 5 6 7 and then
1 again to repeat at the octave. You wouldn't number it
1 1.5 2 3 4 5 6 then
1 to repeat. Numerically it's correct, all the numbers are unique, but it needlessly obscures there are seven degrees total. You would number it with whole numbers.
The same thing basically happens with letters. Seven letters are used for a seven degree scale. One unique letter is used per scale degree. So, if you start with
A♭, the next degree will be some kind of letter
B. Of course we have to spell it specifically as
B♭♭ to get a tone that is one half step above
This system starts to break down when using standard staff and scales that don't use seven degrees. Less than seven isn't a big problem. You just need to decide which letters to omit. But, with more than seven degrees, like an octatonic scale, you must repeat a letter.
You can't avoid these issues. Staff notation was developed centuries ago for styles that were much more diatonic, less chromatic.
Considering the Dorian ♭2 on C based on the second degree of B♭ minor:
X: 0 K: C L: 1/4 _B, C _D _E F G A _B C _D _E F G A _B c
You're correct, as the result is that A♭ it is based on G♭, which actually is B♭♭ major:
X: 1 K: C L: 1/4 _G _A __B _c _d _e f _g _A __B _c _d _e f _g _a
But I wouldn't really suggest so. If you can, use the enharmonic G# (based on F# minor).
X: 2 K: C L: 1/4 ^G A B ^c ^d ^e ^f ^g A B ^c ^d ^e ^f ^g a