Can the socket in my recessed light be replaced without replacing the entire framing kit?
Good question. Yes, it's likely that you have to replace everything. This is one of the downsides of recessed lighting... most of the time the wires from the service tie-in are contained inside a housing or inside of a conduit. It's not weird at all to have it be like that... there are very few can light fixtures where this is not the case (due to fire concerns). 

Can you access the fixture from above? If so, it's easy to remove. It's a little more complicated if there's no attic access to the fixture --- although there should be, because that fixture does not seem to be suited for insulated cavities. If that fixture has insulation touching it, you want to replace it (and all your others where they touch insulation) regardless of condition with an IC-rated fixture.