do i need to wait till 15th March 2012 to start claiming CB and then back bate claim for the last three month (from 15th dec - 15th march)?
You can claim now, but of course need to answer that question, saying when you ceased to be subject to immigration control .... 15.12.11.
Ordinarily a claim might be backdated up to 3 months, but in your circumstances payment will not be back further than 15.12.11.
But certainly go ahead and claim now. It will take a few weeks for the claim to be agreed.
Student fee? My understanding is that your status at the start of the course counts for the whole duration of the course, even if your status changes during the course.
In September 2011 you did not have ILR, so that is the fact that will count for the whole course.