Your making this harder than it really is.

After removing the nuts on the firewall side and the heater hoses(If attached) go underneath slowly pull the box out(So the heater core tubes clear firewall) and down.
Once the box is down on the floor you'll see what need s to be removed next heater control cables, etc,

Then take the box outside of the car remove heater core install new.
Rehook heater control cables etc, then proceed to put the box back up and in.

BE very careful when putting heater core tubes thru firewall as you can damage the heater core causing it to leak.
So look at where the holes are at and keep that in mind as you slowly reinstall the box.