Yes, everyone pays (with the possible exception of the self-employed. I don't know if we pay into this thing or not. If we're covered, I'm taking a 6-week paid vacation each year, courtesy of this system.)
This leave is for more than maternity. It could be for adoption, a death in the family, etc. The employer is supposed to be the 'gatekeeper' of what constitutes an emergency. I'm betting that most employers, since they aren't the ones paying directly, will be somewhat lax.
I don't think you could characterize this as a redistribution scheme just yet. The benefits are capped at some level, and everyone pays a fixed amount into the system. However, when the inevitable abuse runs the cost up, it'll become too expensive for lower-income workers to pay, so I expect that we'll see "the rich" socked for more. That's the point at which point it'll take on redistributionist characteristics.
That's a good point as well, Davis has already been raiding the various California funds by the simple expedient of declaring them to be "in surplus," and taking the money. Then some time later it's "discovered" that the fund is short of money. So the state gives it back. JUST KIDDING! Davis naturally says that to make up for the shortage, taxes must be raised to replenish the fund.
What do you want to bet that this giant pot of money is going to get the same treatment?