Car runs great except it has a rattle in the mid range ?

Does it sound like a valve or piston, or something loose on the exhaust pipe ?

Sounds like a bad rod bearing, no cure except a rebuild. I assume you do know the difference between spark knock and bearing knock.

Not sure on model but to me sounds like its a dodgey valve lifter when the seals start to go they cannot hold the oil and when there under pressure they basically empty them selves hence when you turn the engine off and restart you get a rattle sound untill the valve fills again as long as the cars running ok leave it untill you have to replace them unless there easy to do sone are sone are a pain

Rattle is cat shield shaking because the engine is running rough.start with the obvious replace the sparkplugs,but problem is probably the EGR valve stuck open,buy one then look on the engine to find like part ,got mine @ orielys

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.

Possibly the fuel pump,if the fuel filter is getting dirty it will make the pump work really hard and make noise,I would change the filter.

check your motor mounts also your catlic converter shield

a knocking sound in an engine is never good. with a high mileage engine things start to get worn. so bearing clearances get larger and things begin to knock and rattle. i would try running a higher viscosity oil than is recommended it might help with the knock. but truthfully it probably needs to be rebuild or a whole new motor installed

sounds like a clogged exhaust system , from the outside of the car does the rattle sound like it is coming from the convertor or muffler? if so run an exhaust system back pressure test via a oxygen sensor port or drop the exhaust and run car with open exhaust and see if the power comes back. doing this will likely set a dtc so be prepared to clear codes after test.

Sounds like you need a new exhaust.