Is it really possible for someone on a dependent visa to start working in the U.S.? What does it take?

Yes, if the H-1B spouse has an approved I-140, the H-4 holder can apply for an EAD to work legally.

Yes, L-2 spouses now receive automatic work authorization and can work without needing an EAD

No, F-2 visa holders cannot work under any circumstances in the U.S

Dependent visa holders can switch to a work visa like H-1B, O-1, or TN if they receive a job offer and sponsorship