Do the Jinn have anything to do with the disappearance of objects?
Jinn are one of Allah creatures, and whosoever lost anything should look for it and never accuse niether the Jinn nor humans of stealing it. And Allah Knows Best.
Is it permissible to offer Istikhara prayer (seeking guidance) after Witr Prayer (A confirmed Sunnah which is offered after Isha prayer and before Fajir prayer with a minimum of 3 Rak`as and a maximum of 11)?
It is permissible to offer Istikhara before and after Witr prayer as it is permissible at any time except that in which prayer is disliked. And Allah Knows Best.
Is prayer mandatory on a dying person who goes through a state of unconsciousness, then wakes up?
If he wakes up and was conscious, then he should pray the way/manner he could, but if he wasn't, then he is exempted from obligatory prayer as Allah Says in this regard (What means): "On no soul doth God Place a burden greater than it can bear." [Al-Baqarah/286].
Is the marriage contract considered valid if concluded at home by the marriage official (Ma`zon)?
Yes, it is valid as long as it is registered at the court.