I don't like your chances on the stop sign one.
The CA Vehicle Code requires that you have proof of CURRENT insurance at all times -- see CVC section 16020(a) linked below . But it's a "fixit" ticket, and if you show up at the hearing with proper proof, you'll get this one dismissed, or you can send in proof by mail and get it dismissed that way. See CVC 16028 linked below.
And yes, whether you made a full stop at the stop sign is subjective, but the cops almost always win those close calls if you try to fight and they show up and testify about what they say they saw.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
1) If your insurance was valid at the time of the stop, take proof to court with you to the court date and the judge will dismiss it. There may be a minor correction fee imposed.
2) Usually police officers look for the rebound. When you apply your brakes, the front end of the car will dip downward. After the forward momentum stops, the shock absorbers and springs will bring the car up to the usual ride height.
You may want to invest twenty bucks in a copy of Fight Your Ticket by Nolo Press. It's a pretty good guide to traffic court and is available in most bookstores.
You may also be eligible for traffic school. It's more expensive than just paying the ticket --- because you have to pay the fine and an additional administrative fee, in addition to the cost of the traffic school -- but it will be cheaper in the long term because it won't go on your DMV record, so your insurance rates won't go up as a result.
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like Avvo, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.
1. Just bring proof with you to court.
2. The stop sign is not subjective in the judge's mind - either you stopped or didn't. You can do a trial by declaration first, and if you lose, then you can do a court trial (it's a second chance).
There's a good book by nolo press, how to fight your ticket that can help with the stop sign.
Good luck with your situation. The above is general information and not advice.