AS is easier to cancel out of than AAS. You can cancel out at any point during -S in AS (when Eir hits with the Book) by dashing. AAS, however, requires you to eat the animation delay of the final -S (the Wand uppercut) before you can dash out. The delay between A and S in AS is above an acceptable level. Most times, you'll be at the receiving end of a combo or will miss completely if you are not adept at using AS.
BTW, shifting is both easy and hard for an Eir. With the standard AS launcher, Eir has, quite possibly, the easiest time with shifting out of any of the classes. Shifting with AAS, however, is another story. It is possible to shift with the AAS launcher, but one needs to be adept at either positioning or half-dashing to regain the distance lost during the shift. For example, AASAAAA shift AAAA is extremely hard to perform (unless you are precisely positioned). AASAAAA shift AA dash AS or AASAAAA shift AA halfdash AAAA shift... etc etc. is possible.
Personally, I don't find much use for S-space. It's slow (atleast, slower than just dashing away). A little known trick to spin-dash (as utilitized in the one kLunia video) is to press a direction opposite the one you are facing while you press S Space. This causes you to do a full 180 degree turn (a.k.a spin-dash). This also works for a 180 degree arc opposite the direction you are facing. For example, if you are facing 4, pressing 2, 3, 6, 8, or 9 would cause you to do a "spindash" animation. Only, 6, however, would produce the full 180 degree turn.
About Moonlight Chain... Personally, I find that Chain, as used for the generic Eir combo (AASAAAA dash AS Holy Spirit [hit wall] Moonlight Chain), is terrible. Assuming PvP, at that point, Chain would be hitting for approximately 8-9 damage a hit (x8 = 64-72 damage). The total crapiness is compounded even further by the fact that Chain costs 93 MP at level 1 (i.e. a huge chunk of MP for such low damage). It may help marginally to keep enemies juggled so you can wait for skill CD, but even then, the benefits are only slight since the duration of Chain is only around ~3 seconds.
Dodge is... an amazing skill. It has its uses for getting out of the initial melee strings but its much better if used offensively. To continue where you left off...
At level 7, you can AA Dodge turn AA dash AS... (continued). At level 8 and over, you can do AA Dodge turn [Full melee string]. At level 7 and up, you will not have to prematurely cancel with self-Heal.
Sleepbind combo may be Eir's cheapest combo, but it hardly counts as "cheap," in the general sense of the word. For Sleep->Bind->Fog->Pieces, it costs a hefty 700 MP or so. Compare this to Dainn's Meteor or Sieg's Nado->DFist, and you would find it hard to consider the SleepBind combo as one of the top damage combos.