You'd probably have gotten a quicker answer if you had posted under either Health Insurance or Fringe Benefits, instead of under Senior and Life Settlements.
Who does the letter get addressed to?
I did a couple of these letters earlier today. I addressed them to the student, care of the parent who is my employee (i.e.) Mr. Christopher NoLongerStudent, c/o Mr. John Employee).
Do they pay an individual rate or a family rate if they elect the cobra?
The former student (or, more likely, his parent

) pays the individual rate for COBRA. Don't forget to change the parent's plan if you are three or four tier.
Is it an entirely new plan or do they hop back on the employee's plan as a dependent?
The whole point is that they are no longer eligible to be a dependent. They will be listed separately on your bill. However, they will receive exactly the same coverage as before, just under their own name instead of their parents'.
I'll be happy to help you with any more questions you may have, as this can be very confusing. But please, try to post future questions on the correct forum, okay?

It's sheer chance that I saw this one at all.