As to the employer accusing you of presenting a false doctor note, you say they told you to not return until you had a real note. That does not sound like termination, but a request for a new note. Go to your doctor, get the doctor to fax the note directly to the work fax number and then also bring it into work. If there concern was really about the authenticity of the note, that should resolve the problem. If the reason was not really the authenticity of the note, that will also reveal that there is an ulterior motive.
As to your mother, without knowing a lot more information, there is no way to suggest solutions to that problem.
Good luck to you.
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Whether your employer can let you go because it believes you supposedly submitted a fake doctor's note depends on several circumstances. If you were out because of a work-related injury, your employer's act may be considered retaliation if it did not really believe that the note was fake. If you were entitled to leave under the Family and Medical Leave Act, the only issue is whether you were out sick, not what your employer thought about the doctor's note.
On the other hand, if neither of those circumstances apply, then your employer could do whatever it wants, however irrational. Ordinarily, your employer could terminate you for conspiring with UFO aliens and you could not do anything about it.
I will only address your first question since that's the only one that's related to employment law. Generally, being terminated for false accusation is not illegal (for a more detailed explanation why, please see: http://www.arkadylaw.com/california-employment-law-and-wrongful-termination-forum.html#/20130111/falsely-accused-of-stealing-at-work-2234315/
However, in your case, that would depend on what your underlying medical condition was. If you had a qualifying disability or serious medical condition and the employer used the "fake note" reason as an excuse to get rid of you for discrmiinatory reason, you might have a legitimate claim for wrongful termination. More facts about your employment and your medical condition are needed in order to determine that.
San Jose Employment Lawyer
I agree with Counsel as to the note issue being a "pretext" (an excuse) for an unlawful firing you may wish to consider whether you have a case for wrongful termination. If that motive was to get rid of you for being in a protected category (based on a medical condition) or for exercising protected rights (state and federal medical leaves) then you may have a case.
I have attached a link to a very helpful website from the California DFEH on "Civil Rights 101" which you can review as well as contact an attorney, many of whom offer free confidential and informative consultation. Do not assume you are not entitled to unemployment in that the standard for that award is subject to an agency decision and appeals are swift and before an administrative judge should you be turned down at first.