Question

Is the work energy theorem for variable force is applicable for both conservative and non-conservative forces or not?

Answer 1

Yes.
If there are non-conservative forces then mechanical energy is not conserved. We
write,
W = W_nc + W_{c =} KE_f - KE_i
where W_c is the work done by conservative forces and W_nc is the work done by non-conservative forces. Since W_c = PE_i - PE_f we have,
W_{nc =} (KE_f - KE_i) + (PE_f - PE_i)
The work done by non-conservative forces is equal to the change in mechanical energy.