E1 in 6 membered rings According to my lecturer, if the geometry prevents planarization of the C-centre than an E1 is impossible. As chair and boat are the only stable conformations of a 6 membered ring of carbons, does this mean that such a ring can never undergo an E1? 

Furthermore, why must the C-centre be planar for E1? I guess it allows for maximal stabilization by hyperconjugation?
Well, the answer to your question, without any complications, is pretty simple 

Whatever can be forced into planarity in order to bear the $sp^2$ carbocation will undergo $E1$, this can be easily achieved in cyclohexane, as it can easily have a twisted conformation which is relatively less stable, but yet can bear the carbocation. Same is not the case with bridgehead carbons.