This is correct:
They were married on the first of April 2014, and divorced five years later.
The difference between
had married and
were married, is that with "were", we are saying they became married - that is, the acquired the adjective of "married" - they went from being not married, to being married.
However, the construction "they had X" implies that X is a verb, which needs an object. So, you could say "They had married their partner" or "they had gotten married" (in which case "gotten" is the verb, and "married" is the object). But "they had married" is not a complete sentence.... Usually.
Brian Hitchcock explains in another answer what else is wrong with the sentence, and makes the point that "they had married" can be correct, depending on context - firstly, in that the object can be implied rather than stated, in which case you are saying
they had married [each other] in 2006.
Secondly, you can parse the sentence differently. I've been thinking of it as being in the past progressive tense, but you can also read the sentence in the past perfect tense, in which case "married" changes from being something they did, to something they have done. In that case,
had married would be entirely correct. This is a less common usage though as, colloquially, marriage is usually spoken of in the past progressive - at least it is in my experience, as an english speaker living in england.
Q1) Both of them are correct, depending on context. However, A sounds much more natural to a native speaker.
Q2) Both, again depending on context. Usually it's a verb, sometimes it's an adjective, very rarely it can be an adverb. Treat it as a verb, unless you're sure it's not.