In this sentence, evin is ev "house" with the second person singular possessive suffix -(I)n, i.e. "your house". It's identical to ev with the genitive case ending -In, i.e. "house's", which is why the mouseover help is confused.
If this sentence were "How many rooms does the house have?" I believe it would just be Ev kaç odalı?.
I used that translation, too and the correction said to add "got". Personally I detest the use of "has got" as a synonym for "has" but I know it is commonly used in some dialects of American English. I do think both should be accepted as correct as the meaning is practically identical.