First, remember that MotS has no target because it does not use the word "choose." To answer, though, MotS does not let you put Darkstar into play to save himself from a global kill like Valar.
Generally speaking, when you initiate an effect, you do the following, in order (let's see if I can do this from memory...):
a. Determine the applicable cost (eg, Confession - are you using the "kneel character" or the "kneel influence" cost option)
b. Check play restrictions - including making sure that all required targets are available and eligible and whether the effect is anticipated to be successful (under the new FAQ 5.0 rule for triggered effects).
c. Identify and apply cost penalties (including modifiers to cost penalties, like Neutral House's reduction of OOH gold penalty)
d. Apply cost modifiers (eg, reducers)
e. Pay the cost
f. Choose targets and initiate effect.
So for the MotS/Darkstar question, you have to identify the character you plan to save as part of checking play restrictions -- well before paying the cost. That means the Darkstar in your hand that you plan to pay the cost with cannot be identified as the character to be saved because it is neither in play, nor in danger of being killed at that point. It does not satisfy the play restrictions for MotS when it is in your hand, so MotS cannot apply to it, even though it will enter play before MotS resolves.
That last, underlined part is also similar to a ruling we got relative to the FAQ 5.0 rule for triggered effects. The ruling says that the target that is ultimately chosen for the effect (in f) must be one that was identified as part of checking play restrictions (in b). This is why, for example, you cannot identify a standing character with a MIL icon when you check the play restrictions to play Distraction and conclude that the effect will resolve successfully, but then choose a different character with the MIL icon that is already kneeling when you actually choose the target for Distraction, making it resolve unsuccessfully.
That last bit answers Hast's question about Asshai Initiate: Even though the target is chosen after costs are paid (e vs. f), the target that you do choose must have been present before the cost was paid (b vs. e).