The word "least", perhaps from getting a lot of use, is flexible in the roles it fills as a part of speech. The American Heritage Dictionary lists it with three definitions, all with approximately the same meaning, but as an adjective, adverb, and noun.
So taking an example:
I earn [the] least out of all of us.
If the "the" is there, "least" acts as a noun, the direct object of what was earned (the smallest amount). If the "the" is omitted, "least" becomes an adverb describing how you earned. They both end up with very close meanings, no one would think very hard about it, and so either one sounds right.
A complicating factor is that "the least", as a phrase, has also come to have adverbial status. I speculate that started with prepositional phrases like "in the least bit" fulfilling an adverbial role, and then wearing down over time. As a result, "the least" usage tends to signify something is very small or insignificant in general. "Least" by itself can also act that way, but can also have a tendency to invite a comparison. In a sentence like
The reply will discuss the one you are [the] least interested in.
it doesn't matter all that much. But sometimes it does matter, such as when a speaker says:
I'm not [the] least surprised that she's leaving.
if you leave in the "the" it sounds more general; it's pretty much a fancy way of saying "I'm not surprised she's leaving." But if you drop the "the" it emphasizes the comparison, and I would start looking around the room for someone less surprised than the speaker (or just realize they are a non-native speaker).
As far as practical usage, "the" seems like the safer course as John Lawler says, unless you are emphasizing a comparison. As a speaker of US English, I would tend to add the "the" in all of your examples except "I'm not [the] least surprised that she's leaving" (for which I would actually say "I'm not the least bit surprised that she's leaving", or even more likely "I'm not at all surprised that she's leaving"). Leaving the "the" off for all the others sounds slightly more stiff and formal.
Best Answer
As you say, "Who are customers with questions advised to speak with?" is equivalent to "With whom are customers with questions advised to speak?". The latter will be usually judged "more correct" by intellectuals, and "more pretentions" among the common men, but both mean exactly the same thing.