英语翻译为什么是Do you know which house that he lives {is living} in?而不是Do you know THAT which house he lives {is living} in?其实哪个对我也不清楚。

Do you know which room he lives {is living} in?
习惯用法，which 本身可以引导宾语从句。再用that多余。

其实这个刚才我已答过，根本不需要that，前面已有which
例：
Do you know which restaurant he is likely to be in?
你知道他会去哪家餐馆吗？
Excuse me. Do you know which line I should take to go to Taipei City Hall?
请问一下，你知道到市*要搭什麽线吗？

宾语从句的that可有可无
但是the house that he lives in， that引导定语从句，不可省

一般的句式是 do you know that +句子,或者do you know +名词.但是which 是代词.代词相当于名词.所以用的句式是后者.

第一个句子是正确的,这是个宾语从句,which本身就可以做宾语从句的连词,所以不需要that.that在宾语从句中通常用于引导后面的内容为陈述事实的句子,比如 Do you know that he lives in this house?你知道他住在这个房间吗?
而且 通常that是可以省略的

Do you know which house that he lives {is living} in?