Legal Question in Employment Law in California

Possibly, in fact probably, because the only reason your job is protected while on STD is that you would be working if your disability was not preventing you from working.

Yes.

in general unless an employee is civil service, in a union, or has a written employment contract, they are an 'at will' employee that can be disciplined or terminated any time for any reason, with or without �cause�, explanation or notice.

Employers have the right to demand evidence of medical reason for any leave.