Do NOT drive it until the issue has been resolved. It will cause catastrophic engine damage.
If the vehicle ran fine prior to this, I would not lean towards a head gasket. If the water runs straight through the cooling system and into the oil pan like that, it has to have a severe internal leak and just topping off the antifreeze shouldn’t cause the leak to increase.
I believe it is most likely a failed water pump or water pump seal as these engines have an internal water pump driven by the timing chain. If the water pump shaft or seal fails, water will enter the crank case and mix directly with the oil. The shafts fail often with this design which I believe is due to the direct force and power delivered through the timing chain. My guess is you happened to catch this at a critical time.
It very well could be a blown head gasket, allowing a water jacket to cross over into an oil port or return. Some vehicles have radiators that serve a dual purpose of cooling oil or transmission fluid as well. They have been known to fail internally allowing the water to mix directly with the oil. But I don’t think your vehicle is equipped with such a radiator. It should cool the antifreeze only. The last item that comes to mind is the timing cover may have seals that need replaced as well if it has ports for antifreeze. As for the water leaking on the outside of the engine, this is likely a separate issue. Try your best to trace it to the source of the leak or ask someone for a hand, pouring water in while the other watches and traces back to the source of the leak.