Legal Question in Business Law in California

I imagine that if the employer requires you to install a device that tracked your off-work movements as a condition of your employment, and the matter is decided in court, that this would be held illegal as an invasion of your privacy. But I'm not aware of any specific laws or court decisions that cover this exact situation (although there may be). And litigating employee rights cases can be expensive and attorneys hard to come by. Your options, unfortunately, may be to either quit, find another job, or refuse to install the device and sue for wrongful termination when they fire you. If you are a public or union employee you may have other rights, consult your union rep.

Not really; the employee can resign. However, the better alternative is probably to accept the tracking as a condition of employment, or of mileage reimbursement, and accept it.

If you are paid for time or mileage, sure they can during the work hours.