青い/白い家 is commonly used and appropriate for a house painted blue or white. We usually don't use 青の/白の家 for the house itself though it is acceptable and understandable. We have an expression 青の時代, which refers to Picasso's Blue Period. However, we only use 緑(midori)の家 or 緑色の家 (I'm not sure there is really a house in such color) for a green house because we have no word 緑い.
I feel both words are abbreviation of '青い色の家'
I hear 青い家 frequently, and no problem to understand it correctly.
I hear 青の家 rarely, so some people will understand it as general blue painted hauses, but some people may feel some special house in blue coloured.
I recommend you to use 青い色の家, if you want to avoid unnecessary misunderstanding.