Reaction of cyclohexene oxide with LDA The question given above appeared in the 2016 Indian National Chemistry Olympiad (INChO). What I think is that the epoxide ring is being opened by a base-catalysed mechanism ($\mathrm{S_N2}$). I proposed the following few steps for this:

The diisopropylamide ion, which is a very strong base, would not be a good leaving group. To obtain the desired products, how could the mechanism then proceed? And if not, then what would be the real mechanism? I did not find any reliable information on the Internet regarding ring opening using LDA.
Because of the steric hindrance of the two iPr groups, LDA is a very poor nucleophile. It is, as you note, a very strong base. The two products observed both arise from deprotonation of cyclohexene oxide.
Deprotonation of one of the methylenes alpha to the epoxide ring opens it to give the allylic alcohol.
Removal of one of the protons on the epoxide ring carbons opens the ring to the enol, which on quench gives cyclohexanone.