I've pondered this idea as an alternative for airdodges a few times, and had quite a lot of thoughts on it, including the prospect of it being multi-directional, taking a jump, and so forth, pretty much everything in the OP. Some more food for thought though:
-Air dashes are partially effected by your character's current momentum. Atempting to immediately jump then air dash sideways, for example, results in a dash with the following trajectory:
This would apply to all dashes and all momentum, and is a feature very common to games that include an air dash. A bonus feature of this mechanic is that it allows wavelands and smooth movement, without unforeseen applications.
-On a per-character basis, air dashes are of different lengths, and may include invincibility or other such traits. Bowser, for example, might get a short super armor dash.
-On a per-character basis, air dashes can be acted-out-of at different times. Characters like Metaknight, for example, might do a barrel roll glide that goes a fairly far distance, but is very slow to act out of.
-Also on a per-character basis, the number of air dashes that can be used before landing or grabbing the ledge, as well as the prospect of it removing their double jump, varies. In addition, some characters might get their airdash back upon being hit, typically ones that are otherwise small and/or grouped with a poor recovery, as with Bowser.
-Pressing the airdash input with no direction results in an air dodge akin to Brawl's, has no effect on the number of jumps one can use, but does deplete one of their uses of an airdash.