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

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