It would be У этого здания, but I think В этом здании is preferred since it is specifying the elevators are actually inside the building. Theoretically у этого здания could imply that the elevators are against the building but not inside, so it's slightly more ambiguous. Typically places/non-animate objects that have things tend to have them в... себе as opposed to у себя, from what I've seen.
I completely agree with you about this paritcular sentence, but I disagree with your animate/non-animate statement. E.g. "У этой машины четыре двери" sounds better to me than "В этой машине четыре двери". I think the actual difference is whether Y is an integral part of X or whether it's some internal feature. E.g. "У этого здания четыре стены" (this building has four walls) but "В этом здании четыре лестницы" (this building has four staircases).
Could you say "In this building are two elevators" or is there a rule I am missing.
Without "there"? That sounds very unnatural to my ear.
"There are two lifts/elevators in this building" or "In this building, there are two lifts/elevators" should be OK.
Okay, The more I learn other languages, the more I find out that I don't even know how to use my own language. Both sentences sound okay to my ears. But it could be because of local short hand.
In principle, I agree with you. However, English is an SVO (subject-verb-object) language, and strongly so. Hence "Two lifts are in his building" is perfectly OK but "In this building are two elevators" is not a standard word order (the verb comes before the subject). It may well be acceptable, but certainly not standard. "There is/are ..." gets your out of this grammatical bind by substituting "there" for a subject.
Sounds to me like that would only be standard if you were to continue it as a list which expresses other things also found in the building, as amenities, for example.
2, 3, 4 use genitive singular. Even if you have 22,222,222,222 of something it will also still be genitive singular because that giant number ends in a two (same for three and four)
"There are two elevators in that building" not accepted 17 Mar 2019. Reported.
The Russian does not even use есть, as in another exercise which used identical structure to translate as a building "having" something.