It depends on what you mean by "run out" but there is probably a leak in the tank.
1 simple cause may be a damage radiator cap or simply low fluid.
Yes, you would put the water in the reservoir.
If it is motor oil in the anti freeze, then it is either a cracked head, cracked engine block, or a head gasket blown. If it is transmission fluid in your antifreeze, then the radiator is leaking from the transmission cooler ----- although usually that would cause antifreeze to be in your transmission fluid.
Probably any normal thing like not enough Radiator Fluid or something is wrong with your radiator or a head gasket leak could cause the radiator fluid loss. Does this Help?
If you put the brake fluid in the overflow reservoir just empty the reservoir and pour in a 50/50 mix of antifreeze and distilled water. If however you put it in the radiator itself, I would recommend you drain the radiator and refill. Brake fluid is corrosive and may damage the water pump seal.
Are you sure it's oil? Most likely is engine coolant from the radiator. Next would be transmission fluid. Or it could be from the reservoir holding the windshield washer fluid. Oil from the front is unlikely.
If the brake fluid reservoir is overfilled, the excess fluid runs through the overflow instead of over the bodywork, which would cause damage to the paint work.
broken radiator
Thermostat stuck closed.
It would be on the coolant reservoir.
Radiator could be plugged or restricted
no the red clay like stuff is transmission fluid in your radiator probably leaking from your transcooler