2 of 2 added 1 character in body

Your first question: It takes 6 moves to mate the black king. The only move is 1.e2-e4. All other white moves would take more than 6 moves... Also after 1.e2-e4 there are a lot of variations to long to post it right here.

The simplest way is to get the details and answer the second question yourself is using the program popeye, see http://sourceforge.net/projects/popeye-chess/

Final remark: There is a well known chess problem with white initial position and black king on h4. How to mate in 3 (white to move...)