You are right that with compound nouns in English the first word qualifies the second. For example, an 'elm tree' is a tree, and the word 'elm' specifies what type of tree it is. Likewise, the word 'living' tells you what kind of room a 'living room' is.
I don't know what question you were asked, so I can't tell you why 'elm tree' was the correct answer, or why 'living room' wasn't accepted as an answer.
'Living' is a gerund, which is a type of noun (substantivo). 'Living room' is a similar type of compound noun to 'swimming pool', in which the gerund ( ing form) qualifies the noun which follows it.
Please ask if that isn't clear.