Smart Hick - my apologies! Upon re-reading your post, I saw that you and I agree on "Mike's and my house." Previously, I had noticed only your response to UIP and wrongly assumed that was your way of saying this sentence. Onward!

Sorry, Smart Hick. What you should be splitting up is "Mike's and my house," ending up with "Mike's house" and "my house," which would be correct. You wouldn't say "I's house," under any circumstances.