Hiya han,
Basically speaking you only need speaking and listening with a mark of at least 4.0 out of 9.0 to reach B1 level equivalent, the other 2 components are no interest to UKVI
Regarding your time frame Novie is right as to say you would have your results about June 1 also your current visa runs out on 3/7/14 so that is the latest date UKVI should receive your application for ILR, the earliest date would be 5/6/14 ie: maximum 28 days before current visa expires.
Hope that helps, good luck...