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Key Signatures with Double Sharps/Flats

Don't confuse yourself unduly, enharmonic notes are there to discuss function, primarily. If you travel around the Circle of Fifths (which is where and how you build key signatures) you don't see A#. The key signature you're looking for is Bb major, which has two flats.
 
Well, you wouldn't write a piece in A#, but I guess it's conceivable that a piece might modulate to A# for a bit, requiring a key sig with double sharps.

Depends on how the modulation is prepared. I can't think offhand of an example of it, but I won't say "impossible." But a "Beach Boys modulation" (up a half step) would just be notated in Bb, as there's no preparation for that kind of modulation.