My 2001 7.3 diesel is leaking radiator fluid quite bad. The engine is staying cool. Does it have a gasket that could be leaking & are there more than 1 gaskets?

A manifold gasket has very little to do with engine cooling so, No!
Firstly check your fluid levels are correct. Don't worry, I've made that mistake!
The next cause of overheating could be the thermostat. A cheap and fairly easy fix to do yourself.
Check you coolant to see if there is oil mixed in it. This will suggest a faulty head gasket.
Run the engine at idle speed with the hood open, until it gets to working temperature. As it does CAREFULLY touch the top radiator hose feeling for it heating up. Also listen and watch for the radiator fan switching on. If the pipe stays cool it is probably the thermostat stuck shut.
If the heat starts getting too high and the fan doesn't 'kick' in, it could be the fan switch.
If the vehicle gets TOO hot turn it OFF and let it cool down.

Diesel oil in with the coolant, in the radiator? Very unlikely. Engine oil in radiator indicates a leakage like head gasket, cracked block or head. Transmission oil in radiator indicates a leaky radiator where the hydrolic fluid is cooled inside the radiator.

you have a problem in the heat gaskets (need replace)

waterpump failure.. thermostat stuck, bad cooling fan(s) or relay..repost..

Are you sure it is oil and not just old corroded dex cool? Just asking because about the only way to get oil in the radiator is through a bad head gasket or cracked head which will usually cause the vehicle to run poorly. Here is a few pics of what dexcool can become.

SOUND LIKE COOLANT FANS HAS A FAULTY THERMO SWITCH LOCATED AT THE BOTTOM OF THE RADIATOR ON LEFT HAND SIDE.CHECK THE FUSES 8 IN THE ENGINE COMPARTMENT AND CHECK FUSE 16 IN THE PASSENGER COMPARTMENT FUSE BOX.WHEN ENGINE OVERHEATING IF BOTH COOLING FANS WORKING.OVERHEATING PROBLEM COULD BE BLOWN HEAD GASKET.CHECK ENGINE OIL LOOKS LIKE MILK HEAD GASKET LEAKING.SEE WHEN YOU DRIVING LOW AND HIGH SPEEDS THE COOLING FANS SUPPOSE TO BE RUNNING.IF THEY NOT RUNNING THAT IS YOUR OVER HEATING PROBLEM.

Dexcool does not cause leaks, but it should not be mixed with other antifreeze. Is it possible that the oil in the antifreeze is transmission fluid from a worn out radiator? The radiator cools the engine and the trans, and internally , the radiator may have ruptured?

If your car has the V6, the lower intake manifold gasket could be leaking.

head gasket gone.

Hey assuming that it is a 1hz diesel 99% of the time it is your oil cooler that has corroded through its quite a common problem toyota will charge about $1200 for the repairs and labour unless your game to give it a crack yourself its quite easy its just getting the injector pump off that sucks!!!