No. They can provide your dates of employment and your title. Most will not say whether you are rehireable or if you voluntarily quit. If you worked for the government at any level, these details may be available.
State, county and municipal (city) employees work, respectively, for those distinct divisions of government. All are public employees, however, since all work for government, at some level.
You can't, ever. The employer's record belongs to the employer. If you were union, the employer might have to reveal some of it to the union. If you file an EEOC charge the employer might have to...