Here in the UK Uber drivers have Uber stickers on their cars. I rarely get picked up by a driver without them.
What imaginary other company are they working for that would tolerate that?
It might not be the case in America, but over here it certainly seems most drivers are working for one company.
IMO this is a no-brainer, the answer is yes, double-bill. The same applies if you get two remote jobs, you bill both.
That's definitely illegal.
Maybe fraud and breach of contract if, in the full-time contract, it says "you affirm you do not have another full-time job that will interfere with this job".
Again, I don't do it because I can't handle the stress of two jobs, but I don't begrudge those who can
Additionally, it's a pretty obvious opening to a lawsuit in at least most jurisdictions for breach of fiduciary duty (under the duty of loyalty -- Google this + your own jurisdiction for more details on what it covers where you are) and breach of contract.
Now, the employer probably won't sue or prosecute in most circumstances, so you may get away with it. But that's some high-stakes roulette, especially since you're not going to get UI if you get fired for gross misconduct and you're not going to get a good recommendation.
As a former Lyft/uber driver this is absolutely correct. It gets even hairier: suppose you are sitting on both apps (or even more) waiting for a ride. Do you get to double-bill two companies for minimum wage hours?
I think what is going happen is if Uber and Lyft are forced to recategorize as employees, they get to do something like "compel drivers to wear a uniform", so like a polo with the brand on it, and prohibit wearing of competitors logo. Or prohibit displaying competitor logo on the car (displaying is a legal requirement in many jurisdictions). In the end the take-home for the driver is going to be worse.
I'm quite frankly surprised that Uber didn't preempt the legislation by creating a class of driver that is an employee, putting these sorts of onerous restrictions on the driver, plus other ones like "you must start and end at central processing center, drive an uber-owned car", "requiring shifts on ADA-compliant vehicles", "being required to comply with an uber-generated shift schedule", in exchange for bare minimum wage and benefits.