In bridge, what level of strength do different systems require to bid a short minor?
1)
Yes, there are systems that that open these ugly 11 counts (Repeat after me: Subtract a point for 4-3-3-3 distribution!), but they are (a) strong club systems that limit an opening bid to 15 or so HCP; (b) systems with a weak NT; or (c) systems with a strong club AND a weak NT.

N.B. The average playing potential, in both NT and a suit contract, is almost a full trick less for a 4-3-3-3 compared to a 4-4-3-2 hand. This is evaluated by awarding a point for the doubleton in 4-4-3-2, AND subtracting a point for the 4-3-3-3. You are significantly overbidding your 4-3-3-3's if you do not make this adjustment.

2) NO, not really; but there are systems that say you really only have 12 points (or even 11) because the hand is crap.

3) No, the answer wouldn't change on reversing the minor suits in any system I know of. 

Update from comments:  


Considering AKJ-Axxx-xxx-Qxx: Nominally 14; -1 (flat); -1/2 (poor
honour placement) = 12.5. I would open, in preference order: 1NT if
weak; 1H if playing 4-card majors; else 1C. With proper evaluation of
distribution and honour-support, 12 point hands are worth an opening.
I intend to rebid 1NT over partners 1D or 1S, or raise 1H to 2H. If
partner jump-shifts I will give a single raise. I will not
voluntarily compete past 2S unless partner shows 10+.
Considering AKJ-Axxx-xx-Jxxx: I wold prefer to see the SK trade with
a small clubs, but this will do. Nominally 13, +1 for diamond
doubleton, -1/2 for poor honour placement = 13.5. My preference for
openings remains identical to the previous comment: iNT, or 1H, or 1C
with the same proviso. 


Update #2 - Advanced Hand Evaluation
Those interested in more precise and more accurate hand evaluation could do worse than to adopt the Banzai Point Count. This uses a 5-4-3-2-1 system and then multiples the result by 2/3, rounding a remainder of 1 down and a remainder of 1 up. This can be done independently of any system, or even partnership understanding, to simply improve one's judgement of hand worth. (Of course, if you DO discuss it with partner, it becomes a partnership understanding and must be revealed on your convention card.)