Previous threads suggested PlayIdle() and SendAnimationEvent(). My experience is that the first function doesn't work and the second function works on a very small set of animations (i.e. "BleedOutStart"). It's possible that I am using the wrong names for these animations, but my impression is that they are simply the ones on the list of Idles found in the game (i.e. creating an Idle property in a script and the list of names is there).
Both, PlayIdle and SendAnimationEvent can trigger exactly the same animations.
The latter refers to names of Anim Events directly, and former refers to IDs in Idle Animations window which (these IDs) refer to the same Anim Events.
The only difference between the two is that PlayIdle makes additional checks for conditions that are set for specific Idle in Idle animations window. So it may be that SAE will trigger certain Anim Event, but PlayIdle won't because of those conditions. And this can be changed by creating a new ID of Idle and assigning that Anim Event to it, but without those conditions. Then PlayIdle will be able to trigger the same Anim Event as SAE function.
In short: there's no such Anim Event that can be triggered by SAE and couldn't be played by PlayIdle function.
The case why both these functions can't trigger certain animations everytime lies elsewhere - in behavior files. These files determine conditions under which particular animations can be triggered - for instance: "sitting" anims while character uses furniture, casting anims while char is in combat mode with spells equipped, combat anims where char has its weapon drawn, etc.
With that said, has anyone here successfully forced an actor to perform a wider range of animations?
As for today, the only mod released that allows to do that is http://skyrim.nexusmods.com/downloads/file.php?id=11811, but bear in mind that if you create your own behavior files then your mod won't be compatible at all with other mods that do the same (currently there's only FNIS anyway).