Why did I get a homework problem set for 229B?
It was a typo. You got the right one for 229C.
What am I supposed to do with hbar and c?
The problems assume the natural unit, where both hbar and c are 1. Whenever you work out numbers, check if the unit is right. If it is not, you can use arbitrary powers of hbar and c to make it work.
In the problem 1(1), what kind of final states shall I think about?
Consider 2 to 2 processes. They are called "quasi-elastic" processes.
I seem to find only 8 events from SN1987A neutrinos rather than 12 events in the paper I looked up.
You are looking at the wrong paper, Bionta et al (IMB Collaboration). Make sure that you use Hirata et al (Kamiokande Collaboration).
How do we deal with the fact that we don't know well when the supernova exploded from optical measurement, which was first seen only a few hours later by luck?
You don't deal with it. The only thing that matters to this problem is the spread in the neutrino arrival times, not the absolute time. The absolute time in neutrino arrival and "photon" arrival matters, however, when you test the equivalence principle between neutrinos and photons in gravitational field.