1. Does the heater core always have water flowing through it, or is that dictated by the heater control valve in-line?

2. How much of a nightmare is it to replace the core?

mechanical flow control valve on front wheel house. wd40 or similar will free it up. when it's too stiff ya end up bending the control cable, fixable.

haven't but don't think it's any worse than the instrument cluster, check the fsm.

haven't done a late b.
done an early and 66 and 67, plus a couple of c-bodies.

valve does block flow. but not completely on its own. some of the back pressure from flowing into the heater core stops it up as well.
so if it is leaking, even with the valve shut, it will leak some.
most people just loop the heater hoses back on them selves to prevent further wet carpet.

getting it out of those others was a pain. it barely fits between the floor and dash to get them out. having to hang under it also sucks, pulling bolts and such.
it is easier with the glove box out, but not necessary.
the fresh air vent j hooks and finding the other hangers on the box holding it up were probably 2nd worse.
worst was getting it out from under the dash once you got it down.
I did unbolt the lower bolt holding the dash in so I could pry it back and away to give me just a little extra clearance to get the box free.
people say pulling the seat makes it easier too, but I never did that either.

then you just have to split the box open to get the heater core out.
most of the gaskets will turn to dust on you when you do that. so be prepared to have new ones.

Can’t be worse than the cluster

Happily, no wet carpet, just steam from the vents and dash area. Control valve is free. Will just do a hose bypass for the moment, looks like new cores haven’t been made since 2017, none available. Found my winter project!

Had that happen to Warhammer. Pulled the box and thoroughly cleaned it out so get rid of the coolant smell.