Maybe, maybe not. Employment law is just not that simple. There is no way to completely answer your question without more information regarding your location, the length of time spent with each employer, the reason for termination of employment with your previous employer, and the reason you want to quit your new job. You need to tell an attorney your entire situation to get an accurate answer.
Absolutely
Unless there were extremely unusual issues in your separation, no. You can file on employers you worked for in the previous 15 months in most jurisdictions.
The employer pays its unemployment taxes to the state the employer is located in. You might file your claim with the state you live in, but your state would then process the claim through the "liable state".
They may ask you, and they may ask your previous employers. However, in certain states, if your previous employer is asked why you left they can only say that you quit or you were fired, they cannot discuss the reasons why. Unless, of course, you violate your non-disparagement agreement you may have signed.
You have to apply for it. You can only receive unemployment if you are currently unemployed, searching for a job and otherwise qualify for unemployment benefits. Some of the qualifications include not being fired for any cause of your own, having worked at your previous place of employment a certain # of hours a week for a certain amount of time, etc.
No. You can only collect from the "liable state" which the employer pays unemployment taxes to, which in your case is California.
In most states, unemployment information is confidential and known only to the state office, the claimant and any employers directly involved in the claim process. To find out your own state's position, contact your unemployment office for clarification.
After you have filed your claim, the state's investigator will contact the employer to get their version of your application. After that, the state will notify you of the next step in their procedure.
legally By filing an unemployment claim and if the state finds for the employer you can appeal the state's decision.
This would depend on the basis for the disciplinary action. Under your own state's unemployment security laws the employer is allowed to discharge a worker for a variety of reasons. If those reasons are allowed and proved by the employer, the unemployment office would find for the employer and you would not collect.
An employer may do any number of things, but that does not mean its legal, ethical, or allowed by the state's unemployment office. When you file your claim, the investigator will check out your story with the employer and vice versa to determine the truth of the matter.
Until you have worked for the new employer for one calendar quarter, the state unemployment commission does not know you are employed there - the employer has not yet paid UI taxes associated with your name and SSAN. Any UI claim will be charged against the former employer ... or just denied.