Overheating, heater core?

I would do a check for a bad head gasket test first. Even a clogged heater core wouldn't cause overheating. However, a head gasket leak could easily cause the overheating.
https://www.2carpros.com/articles/head-gasket-blown-test

Well, it started with the heater blowing only cold air and overheating then it would straighten up off and on and I noticed a leaky water pump from the weep hole then lead to a cracked radiator followed by a cracked lower intake manifold so did the thermostat and vacuum lines cause they were brittle and cracked in the process now the heat works as long as I'm moving but if I sit for a bit it starts blowing cold air and temp gauge rises but I start moving again and the heater starts working and the temp gauge drops slowly back towards normal. No oil in the coolant or coolant in the oil, no smoke or liquid in the exhaust, does build pressure a little quick in the radiator though and disappearing coolant. So, I'll pull the plugs and go that route first, I guess. Wish me luck and any ideas are always welcome.

There is a quick test you can try, take a cheap balloon and put it over the coolant fill. Start the engine and rev it a bit. If the balloon inflates rapidly, you have combustion gases in the coolant.

Well you were right, head gasket is blown. Builds pressure was too quick, bubbles in the coolant, you were right on. It's crazy the side effects I mean it blew out the radiator and cracked the lower intake and etc. But I am doing the head gaskets now luckily, they are not cracked or warped themselves truthfully. But I ordered this head gasket set from carparts.com it was only $49.99 so score! But anyway, it came with a few gaskets and o-rings and these 10 plastic things I'm not sure what exactly they are. Can you possibly help me identify the different parts in the set and what they might go to? I mean I know the ones for the heads, the upper and lower intake manifold and the EGR valve, the throttle body, thermostat, but not sure about the rest of these.

Most head gasket sets include the valve stem seals (look like O-rings) they will have things like thermostat housings, water pump, exhaust manifold gaskets, crossover tube gaskets. Some of them you won't use unless you strip the head down and rebuild it.