Sometimes. The Federal Bureau of Investigation's (FBI) National Crime Information Center (NCIC) criminal records database may be accessed for some criminal history checks, depending on the kind of job involved. The National Child Protection Act authorizes states to establish procedures for criminal history checks using the NCIC  for employees and volunteers who work with children, elderly individuals and persons with disabilities.

Generally yes. Service records can only be released under limited circumstances unless you provide authorization (the military can release your name, rank, salary, duty assignments, awards and duty status without your consent). Any request you don’t authorize would have to be made under the federal Freedom of Information Act.

They shouldn’t, but definitions of expungement vary from state to state. The records might not completely disappear and might still be available to law enforcement. There are also data brokers that collect and publish publicly-available information whose websites may not be updated to reflect an expungement (making it possible for someone to come across an expunged record). 

No, the California Labor Code prohibits an employer from seeking salary history information about an applicant for employment. Salary history information includes both your rate of compensation and information about other benefits. The law applies to all employers (including state and local government). You can voluntarily disclose salary history information to a potential employer.