Coolant in oil

Question edited for clarity and typo 'weephile'.
Question moved from Misc to model category.

Screeching is dry metal on metal abrasion. Take it all apart and get some water pump grease on the shaft or replace it completely.

Seems to be a known issue

https://www.google.com/search?q=2002+Yamaha+YZF-R1%2C+coolant+leaking+from+weep+hole

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A car that keeps overheating should not be driven.

Light blue smoke is burning engine oil. This is probably a sign that overheating has damaged the engine.

Water pump can leak coolant, make a noise and even wreck the engine under some circumstances but they have little to do with engine oil. If the water pump leaks it is an external leak.

If there are no external coolant leaks then yes it could be the head gasket or perhaps a crack somewhere...

I would think maybe the water pump is leaking you will have to put it up on jack stands and wipe everything down under the car. Then if you start it for a minute or two shut it off you should be able to see where it's comming from.

If the water pump is near the starter it is in the
wrong place,it should be on front of engine.(sorry)
You could be loosing water from a pipe above the starter
one which goes to the heater.Although minor,it is leaking
onto starter and you do not want that.You must cover or
put something over starter to stop it getting wet,and this
is URGENT like yesterday. You will short the whole car
if not.
LUCK
Syd

It is a possibility that the leak may come from bad seals from the water pump or a bad water pump. If you also observe bubble in the coolant inside the radiator when the car is running or water coming out of the exhaust you may have a bad head gasket. The coolant comes out of the bad part of the head gasket near the outside which lets water outside the engine as well.

SMOKE UNDER HOOD COULD BE COOLANT LEAK FROM RADIATOR ITSELF OR CHECK FOR LEAKING TOP RADIATOR HOSE AND CHECK FOR LEAKING BOTTOM RADIATOR HOSE.LOOK UNDER THE HOOD IF COOLANT BOILING OUT COOLANT OVERFLOW JUG ENGINE OVERHEATING FROM BAD THERMOSTAT OR BAD WATER PUMP.CHECK FOR COOLANT LEAK AT WATER PUMP WEEP HOLE.IF SO WATER PUMP NEED REPLACING.ADD MORE COOLANT UNTIL LEVEL CORRECT.TRY LOCATE LEAK. IF CAR OVERHEATED AND BOILED IT OUT. SMOKE UNDER THE HOOD CAN ALSO BE CAUSED BY LEAKING VALVE COVERS WHICH CAN BE FIRE HAZARD IF OIL LEAKS ON HOT EXHAUST MANIFOLD.

You said "a sign for engine oil" rather than "a sign of engine oil" so I interpret that to mean that the oil level light (shaped like a can with a spout) in the dash is illuminated?
The oil level light on these cars tends to get hyper sensitive as the cars age. It will start flashing when the car is only a cup or two low on oil, and will eventually light all the time. The sensor can be replaced while you are doing an oil change, or you can just take the bulb out of the dash and ignore it.
If the car is actually consuming oil, and the oil level is actually low soon after a change, that is not good. The first thing you should do when an e46 starts consuming oil is completely dismantle the crankcase ventilation plumbing and clean it Pay particular attention to the 'Y' where the oil from the oil separator drains into the dipstick tube. I think there are instructions for this job on the Bavarian Autosport website. You do not need to remove the intake manifold to do this job (but if you don't, small hands would be handy).
As for the water leak - do you know where the original leak was? The most common leak is the water temperature sensor on the lower radiator hose. Squeeze the electrical plug and pull it off the sensor and see if there is evidence of coolant on the sensor. The problem is that the O-ring around the sensor shrinks and hardens over time, and no one will sell you just the O-ring.
The second most common water leak is a failing plastic radiator. If your radiator is original, it is about due to fail. The top big hose nipple crumbles from the inside and eventually blows off suddenly. This usually occurs at an unfortunate time, like when you are rushing to the airport to catch a plane.
The third most common water leak is a failing water pump. The water starts leaking from a tiny weep hole beneath the water pump. You MIGHT be able to see the bottom of the water pump with a light and a mirror, but usually I cannot see for sure if it is leaking until I get the belts and pulley off.

Low fluid light means check your coolant which prob whats on the ground. Depending on the rate of leakage fill coolant start try and locate leak sounds like a hose or water pump slow leak prob water pump fast leak hose good luck

sounds like a gasket is leaking check your engine oil and make shure no coolant in there ,will float on top of oil and fog sight glass if fitted. do not use bike if so. it may be a long shot but look at the rad cap if also on the left. it may be leaking on the go when hot under pressure , change for piece of mind, money is never wasted on safety features .finally check other side of engine as leak may be running from one side to the other.

The opti-spark distributor cap is below the water pump.
Often a leaking water pump will cause a short in the distributor
Due to the amount of time and labor cost to remove the water pump, and that fact that the water pump has to be removed to replace the distributor cap, it is a good idea to replace the distributor cap at the same time the water pump is replaced. The water pump is driven by a pinion shaft with an " O " ring. Be sure to replace the " O " ring too else you may shortly develop an oil leak at that point.