Do you have to have written authorization from CO for a soldier to do Uber during off duty weekend hours?

Yes. Joint Ethics Regulation 5500-7R.

AR600-20
4–17. Standards of conduct
Department of the Army personnel must place loyalty to country, ethical principles, and law above private gain and other personal interests. The performance of their duties should be in keeping with the highest tradition of military and civilian service to the U.S. Government.
a. Guidance. Minimum standards of conduct required of all Soldiers and Army civilians are prescribed by DOD 5500.7–R that provides Army personnel with guidance on a multitude of ethical issues, including the avoidance of conflicts of interests between their commercial/financial interest and their official duties.
b. Annual training. Commanders at all levels will ensure that all Army personnel required to file either a public or confidential financial disclosure report, contracting officers, procurement officials, and others identified by an Army ethics counselor, receive face-to-face annual ethics training as prescribed by DOD 5500.7–R.

This is likely amplified in a Senior Commander or Installation Policy on how/what to do. Obtaining the permission in writing moves you off the X and covers you as long as you operate within that permission. They counsel you on your duties are first, they will take precedence, and if there is conflict, your second employment loses out.

No. Written authorization from CO for a Soldier to work outside of duty hours is usually needed if there's a chance or strong possibility that the work could interfere with their duties. Uber allows you to set your own time schedule and be your own boss, so there shouldn't be any issues

I would say no, but depends on how much of a stickler the co is and how he interprets the rule. It's not employment, you are a contractor with Uber right? Independent like being in a band and playing bar gigs at night?