2002 Ford Taurus Oil Disappearing?

I bought this car with 35,000 in 2004. Traveled the next day on a trip of about 1,000 miles. By the time I went 500 the oil was almost gone. So began the 'adding oil' every 2 weeks. It went to the repair shop several times, they replaced a few things.

It never fixed the problem and the strangest thing is, the car does not smoke, there are no leaks under the car, the gaskets were checked no leaks.
Where is the oil going?

Also after all this 'losing oil problem' I got the carfax. It showed the car was towed to a station, then to the Ford dealer, then sold to a car auction.

Someone wondered if the car took on water?

It also has other issues (electrical?) Like 4 starters, losing a few minutes of clock time every few weeks, a strong fume smell running the Air or heat.

Is this just the case of a lemon? I paid cash and own the car so I have been trying to keep it up to stay debt free. Any hope?

Thanks for any insight!