The first one is nothing close to correct. :P The second one would be correct if it was "Mine and Michael's bed is >>AS<< hard as a board." Then it will be correct.
This is a good question. (I'm assuming the missing "as" is a typo.)
Re: Michael's and my bed is as hard as a board.
This version is better than the one below, because it is more clear that you are talking about one bed. To make it even clearer, you could say "My and Michael's bed.
Re: Mine and Michael's bed is as hard as a board.
This version could also be talking about one bed, but logically it aligns better to talking about two beds. Remember, "mine" is an absolute possessive pronoun. Here "mine" means "my bed". It does not modify "bed" like an adjective. So, it should be "mine and Michael's bed are..."