How early does the parking lot for Downtown Disney open on weekends? Since I believe some of the restaurants are open for breakfast I assume early, but how early?

I've been there as early as 4:30am, I don't believe anyone is in the booths until after 10am. The time limits still apply 3hrs free 5 hrs with movie or restaurant validation.

The Main Lot is open 24/7.

As mentioned above, the rules apply on parking hours 24/7, though Disney tends to close down the toll booths an hour or two after House of Blues closes, and remains closed until about 2 or 3 hours after Disneyland opens. If you leave during that time, they just let you drive thru (the amount of revenue is just not worth staffing the booths, as most folks fall under the 3 hour rule during that period). But if you show up after they open, be prepared to pay a lot if you want over 3 hours, or 5 hours with validation. I think it is $12 per hour currently.

I think it is $12 per hour currently.

Still $6 per hour actually, which is pretty reasonable compared to other places.