Lots to address here.
OK, first off, you're not going to be able to determine ANYTHING with the engine running. The coolant won't reach operating temp with the cap off, and even if it does, it's no longer a "sealed" system, so you;re gonna get 200* coolant all over yourself and the ground when the thermostat opens.
Coolant is NOT streaming out of the relief tube because it's not pressurized. Gravity and atmospheric pressure (14.7 psi) are kicking your butt.

There's nothing to force the water UP and OUT of the relief tube. Relax.
You're not going to see coolant flow/movement/rushing through the radiator with the cap off, for one of two reasons: Your thermostat isn't opening (because the system's not pressurized by a cap) OR your thermostat is shot.
Last, heater temp at idle is irrelevant to your cooling system. That can be a bad heater core, a bad air mix door, a vacuum leak to the air mix door mechanism, etc. Separate issue. Skip it for now.
Here's what needs to happen: Replace the thermostat. Should be done every few years anyway, just do it. It's cheap. Replace the cap while you're at it. It's probably not operating optimally anyway.
Fill radiator to top of neck. Open the relief port, and top off radiator completely. Close relief port, squeeze upper hose with both hands to "burp" the radiator, top it off one more time and put the cap on. Fill the overflow to 1/4 full, and go drive. You're done. Any air in the system will be evacuated into the overflow and replaced by coolant (when the vehicle cools).
Welcome aboard - GREAT question!