Hello and welcome to Just Answer!
Both heater hoses to the heater core should be hot, with one(outlet) slightly cooler.
The first thing I would do is check to see if the engine is running at operating temperature(around 195). The best way to do this is to use a scan tool and monitor the coolant temp sensor. If the coolant temp sensor shows operating temp, then you should have hot heater hoses! If you don't have access to a scan tool you can use an infrared temp gun to measure the engine temp. Most of the time you can tell by just looking at the temp gauge! If the temp gauge is up to normal operating temp it should throw heat! If the engine is running cold then you will need a thermostat. If the hoses aren't hot, check your coolant level and make sure there is no air in the system. If one hose is hot and the other one alot colder then you will have a stopped up heater core. Sometimes you can back flush a heater core and it will work better, but most of the time the heater core needs to be replaced.
I hope this helps, let me know!
The engine water temperature gauge goes from 40 - 125 at it is running at 100. The radiator inlet hose is very hot, while the hose to the heater is warm, but no where close to the heat of the radiator hose.
I thought about back-flushing since I'm in Seattle and now is not a good time to be working on a car.
The heater core outlet hose appear to have a split/joint half-way to the firewall. Is the best method to disconnect this and attach an outside water hose and slighlty increase water pressure? Also disconnect the inlet hose so the core can drain?
I'll accept your answer but would appreciate a response