Yes, Mirage Mirror does target--any time an ability says the word "target" it is a targeted ability. As such, you cannot use it to copy a shrouded artifact, creature, or land, and can only use it to target a hexproof artifact, creature, or land if you controlled the targetted object. Also, note how it does not use the word "permanent"--Mirage Mirror can't target planeswalkers.
Compare to Phantasmal Image, which enters the battlefield as a copy of another creature, but does not target that creature.
Yes generally, but there's a bit of an error in your terminology. Yes, the Illusionary Servant will die, but it will be sacrificed, not destroyed. These are very different effects, so the distinction is critical. For example, an indestructible Illusionary Servant would still die as a result of Mirage Mirror's targeting, as sacrificing a creature does not implicate indestructible.
October 5, 2018 3:12 p.m. Edited.
@cdkime Thank you for the most detailed answer, truly appreciated!
October 6, 2018 4:15 a.m.
Your Mirage Mirror will not become a copy of the targeted object if it is sacrificed prior to the ability's resolution, whether it is due to the illusion ability or the controller sacrificing it to another effect. Targeting legality is checked on both activation and resolution, it the target is sacrificed, destroyed, given shroud, or otherwise made illegal after you target it, you won't copy it.