What kind of official background checks are available to US companies not working with classified stuff? I guess like, none? Maaybe some states offer this as a service?

It isn't through the government, but the credit bureaus and paycheck processors provide these services. They can also provide salary history.

I work in the trucking industry. There are many private companies that provide full employment history checks, criminal background, driving record, drug and alcohol testing history, etc. Two off the top of my head are Asurint and HireRight.

Yeah, these aren't any kind of "official" background checks though. You're just buying a report of very questionable quality from a data broker.

It's often not very difficult to create false records in these reports.