1996 Volkswagen Jetta Oil Smoke
That may even kill the low coolant light( but don't disconnect it and leave the wire loose) Something is wrong.
Make sure that all the connection tabs off the sensors are shiny clean and that the connector fits on tight and that the wire is not broken at the connector(happens alot on those thin wires). Is the connection grounding out when it shouldn't be or vice versa.? The low oil pressure light MAY be confusing the "check engine sensor and it comes on as a backup warning(both are related to trouble in the engine room. It could be the sensor is faulty, but I would hazard to guess the wiring connections are the problem first. Correct that and see what happens.
With the chatter noise, from your description reminds me of pinging. Try changing the brand of gas and higher octane. It is an easy way to check. Work from there. Maybe a change to cooler plugs would be all that is required.
The pressure leak subject needs some more clarification.
These are suggestions that have worked for me on past project vehicles which I have suffered through.
Step by step you can overcome.
Suggested Repair
Inspect for a failed intake or head gasket that may let oil enter the coolant and leak onto the exhaust. Check the oil level and external oil leaks, since low oil can cause valve or cam-follower tapping, smoke, and a burning-oil smell. Also verify the coolant level and warning-light sensors separately because the coolant light stays on.
Conversation Summary
The owner of a 1996 Volkswagen Jetta reports an oil-pressure warning after about ten minutes of driving, a coolant light that stays on, intermittent check-engine light with power loss, a pressure-leak sound, burning-oil smell, occasional smoke, and valve chatter after driving. The expert suggests a bad intake or head gasket allowing oil into the coolant and onto the exhaust, with low oil reducing valve and cam-follower lubrication. Other replies suggest thicker oil, a faulty oil-pressure sender, electrical connections, or fuel-related pinging.
