Ok, I'll try and answer everything but I'm on my phone so it will probably be filled with typos.
GPA: 4.6 weighted, 4.0 unweighted. I don't know my UC GPA.
SAT 1: 2300 composite (770 writing, 750 math, 780 reading)
SAT II: 800 US History, 740 Bio E, 720 Spanish
AP Scores: English Lang, Bio, Spanish Lang, US History- 5
Leadership: President of Key Club for 2 years, Co-captain of academic olympics
ECs: National Honor Society, academic olympics, key club, speech and debate, mock trial.
Over the summer I took a class on law at Stanford between sophomore and junior years and between junior and senior years I was awarded a scholarship through this thing called the Congressional Academy to study political science for college credit and tour DC.
To answer your questions:
1) No. Around February (I think) they send out invitations to apply for the scholarships to the applicants they deem are in the top 1% of the pool. Then, you must submit an essay and a letter of recommendation online as your application. I was informed that I won right around the time admissions came out.
2) I think like the admissions process itself the selection of Regents Scholars is holistic and somewhat mysterious. I'm pretty sure battle with financial difficulties could be a benefit in your application but by no means do they select only those with enormous need. Personally, I had an EFC of 817 so the scholarship covers everything but $817 of the expected cost (which I'm covering through an outside scholarship). But if you have no need you are guaranteed $2500. Plus you get the perks of guaranteed housing and parking all four years and priority enrollment.
Let me know if you want to know anything else, I'm happy to help