You should remove the entire heater box to service the heater core.
Yes. I tried to remove the exposed interior half of the heater on my 66 back in 1967 without removing the entire unit and it didn't work well back then.
I highly recommend you remove the passenger seat. You'll regret it if you don't.
Your back will thank you.
Remove glove box.
I've never thought this was necessary. If you do take out the glove box, it is a lot easier to remove and install the two control cable retainer nuts (351699-S, available AMK) on top of the heater box. If you have a console, remove that.
I do recommend you getting a new heater hose. "Loop" it thru the holes in the firewall without cutting it to fit (yet) then attach it to the heater core fittings with the correct clamps. Install the heater box to the firewall. Lay the looped hose in position placing the portion from the lower heater core fitting to the choke hose bracket on the carburetor (ex HP engines) then to the manifold fitting for a
length sanity check. The
sanity part is - are you going to have enough hose from the top fitting to reach the fitting on the water pump? If yes, cut to length (I cut it three times and it was still too short). If possible, start the engine and check for leaks in your installation. If none, reassemble the interior.
Jim