I think the heating core is shot

You have either a leak in a heater hose or a leak in the hearer core. Fill the radiator and turn the engine on. Let get up to operating temperature. Now you should be able to see the leak. You can take it from there. If it is the heater core, you will probably want to let a mechanic replaoe that.

Sounds like you have overheated the engine.

sounds like maybe your engine is low on oil.

Sounds more like an "engine running poorly, needs immediate attention" problem. I would first try to isolate that "terrible rattle" sound. Does it do that at every start-up? or just started? I'm just wondering if it's your starter noisy, or something internal in the engine. Have somebody listen to it when starting and see if you can find it. That is most worrisome. The clicking of the starter is very common to toyotas: the starter itself is almost indestructible, but the solenoid attached to it often needs servicing because the contacts inside tend to wear.
The dieseling you speak of is sometimes caused by too much fuel left in intake after shutting off. Engine keeps turning over and firing from hot carbon deposits inside your cylinders igniting the unburnt fuel. It could be from a rich running condition. Buy a can of liquid Seafoam. 1/3 in gas tank, 1/3 in oil crankcase, and 1/3 poured slowly into intake with car running at a fast idle (may need to bump gas a little to keep it from dying: keep your engine running. May smoke a lot, but its just your engine cleaning out, loosening up the carbon and burning it off. It will keep smoking for awhile, but after a few moments driving, it should begin to clear up. If your engine is in good shape, you may notice the difference immediately in better power smoother performance.

Your heater core is shot. Coolant leaking on the floor (wet) and driping onto exhaust (smoke).
Replacement of the heater core is the only proper fix. As a temporary measure , you could disconnect the hoses that run to it , and plug them (or join them together). Engine will run , but you will have NO heat in car. enough of your coolant has probably leaked out to cause engine problems. Make sure it is refilled and air removed from system before driving.

Your heater core is leaking. if it was left disconnected the engine coolant would have come out immediately upon starting the engine the very first time..

You need to replace the water heater core

bypass the heater core then check the water pump to make sure it is still working

I hope that you had the proper weight oil installed, at your oil change. If your oil is up to the proper level, and it's rattling ,this is VERY bad. White smoke usually means steam in the exhaust, and that means a blown head gasket. Check your coolant level. I wouldn't drive it anymore......until someone knowledgeable can diagnose it!!!!!

The heater in your car depends upon a flow of hot coolant going through the heater core in order to get hot air into the car. If the heater core is partially plugged, it will restrict the flow. If you have a head gasket that is putting exhaust into the cooling system, it will eventually get into the heater core. When this happens there is no coolant in the heater core to make the air warm. If the water pump vanes have corroded away the result will be very poor coolant circulation causing the heater to not blow hot air. If your car is equipped with a heater control valve that is not functioning properly, the coolant flow will be restricted. One or a combination of these factors is likely the culprit in your case.

if the car is not overheating, the thermostat is working fine.... replace or flush the heater core multiple times to try and eliminate sediment build-up.