What is the relation between adiabatic elimination and adiabatic theorem? Does adiabatic elimination come from adiabatic theorem?
Yes, the adiabatic elimination is a particular application of the adiabatic theorem in a more complicated context.
The adiabatic theorem says that if the Hamiltonian is changing as a function of time very slowly and there is a gap in the spectrum around the eigenvalue $E$, the eigenstate with this eigenvalue will continuously evolve into the "same" eigenstate of the gradually changing Hamiltonian, without getting a contribution from other eigenstates.
Adiabatic elimination is a "practical" application in systems with at least three states. The first-to-second transition may be driven by an AC field and the second-to-first decay may be eliminated by the effect described in the adiabatic theorem. The time-dependent Hamiltonian in this case involves the gradual increase of the Stark shift.