Temp sensor goes just under your thermostat housing (between the outlet and the upper timing cover. Just under the bye-pass hose here...
The heater core are the two small black hoses on the firewall just to the left and right of the back of the valve cover. Seen on the left side of this picture.
The connect to the heater core and go though the firewall. I would leave them in place and flush the entire system not just the heater core.