"In the past two years, have you been employed by a company with whom you were subject to DOT required drug and alcohol testing?
I know we had preemployment drug screening and are subject to random drug screening at 121 carriers, but I'm not sure if that's a DOT requirement or company/FAA requirement.
Could anyone verify that for me please?
Thank you.

All 121 (and all or most 135) are subject to the DOT testing.

The FARs simply refer you to the DOT rules for transportation workers, so there's no separate FAA system.