"Whom The Lord Calls, He Qualifies"?

I commented (May 2008):

It is often said in the LDS church that everyone who is given a particular calling has a "mantle" that falls upon him or her that qualifies him or her for the calling. This premise is utilized as one reason why members (both those called and others) are told not to doubt the divinity and accuracy of the calling.

In my own life experience, however, it appears quite obvious that in many instances (probably most), people are given callings based upon the plain-view observations/perceptions of their leaders. Some people may have hidden weaknesses or faults which if known to the leaders during the deliberation/inspiration process would make the leaders select someone else for the calling.

My questions are these:

1) What does it mean to say that a person is "qualified"? Does it mean that they are necessarily worthy in God's eyes? What about the occasional child molester who gets called to be in the bishopric? Is that person worthy? Would they have been called had that serious fault been known?

2) Is it universally true that every calling is divinely inspired? If not, doesn't this undermine the idea that church members are required to never criticize church leaders?

3) If not all callings are divinely inspired, how do we know who was truly called by God and who was not? (i.e., what if an apostle were called because of family connections or politics, rather than God's specific desire that said apostle lead the church one day?)