Learning to fly is both harder and easier than you think.  Using the controls to fly the plane is surprisingly straightforward, and you'll pick up basic skills very quickly.  On the other hand, flying safely and responsibly means knowing quite a bit about aerodynamics, aircraft systems, aviation rules and regulations, weather, and a variety of other topics.  Mastering the breadth of knowledge required to be a safe and competent pilot will require focused effort on the part of the student.

If you can make 2-3 training flights a week, and put in at least 3 hours of home study each week as well, getting your license will take about 3 to 5 months.

It differs for each student, but you can expect to spend about $10,000 at today's prices.  The largest single expense in flight training is the cost of renting the aircraft you'll be training in.

On average, a student pilot in the United States trains for about 60-70 hours in the airplane.

Some people finish Ground School first, and then begin flight training after they have developed a pretty good working knowledge of aviation issues.

Other people choose to do Flight Training and Ground School at the same time, and find that each reinforces the other.

You can start this Tuesday!  The Ground School is broken into 12 separate lessons that cover different areas of knowledge that are remarkably distinct.  New students can begin at almost any time.  Just start on whatever lesson happens to be that night, attend the class through the end of the cycle, and then continue attending the next cycle until you have been to every lesson.  Probably about 80% of students start mid-course in this way.

If you want to wait and start on Lesson 1, that's okay too.

I'm not sure.  My understanding is that the GI Bill will only reimburse training done at an accredited education organization, like a college or university.  Most of the flight schools in the Bay Area are certified under 14 CFR Part 61 which covers independent flight instructors.  Larger, more formal and structured flight schools are certified under 14 CFR Part 141 and require a pre-approved syllabus, instructor training, and methodology, and other factors associated with accreditation.

There may be Part 141 schools in the Bay Area that I don't know about.  I suggest you check the FAA website and get recommendations for flight schools.  If that doesn't work, you could simply call other schools in the area to ask if they take GI Bill.

Embry-Riddle is a university program that teaches flying, with campuses in Prescott, AZ and Daytona Beach, FL.  My guess is that they will take GI Bill stuff, but it may have to be incorporated with an undergraduate degree.