Hmm, while I think that "we are happy with the rain" is nearly functionally identical in meaning to "we are happy about the rain," I think that the latter is more appropriate here. Firsty, because it is a logical extension of the phrase "we're happy about it;" and secondly, because the word "über" in this context translates to "about." (Native speakers, please correct me if I am wrong).
This is a reflexive verb. Check it: http://german.about.com/library/weekly/aa022601a.htm
English has the same construction for a similar phrase: "Did you enjoy yourself?" That's not materially different from "Did you enjoy?"
An interesting question I thought. When "über" is used to mean "about" (rather than "over" or "above") it always takes the accusative even though, as there is no motion involved in this case, you would normally think it should take dative here. This is therefore an exception to the usual rules for "two-way" prepositions.
Yes, it is intentional because here we have a reflexive verb "sich freuen" = to be happy, to be pleased. It is a bit confusing because the grammatical structures are very different.
check these for further explanations and examples: