What is/are the product when this reacts with Lucas reagent? What would be the product(s) if this alcohol reacts with Lucas reagent? 
Would Methyl shift happen ? I heard that in Groove's process, only primary and secondary alkyl chlorides are formed.
A shift of the methyl group is likely, because this would yield a more stable tertiary carbocation. In concentrated hydrochloric acid, the double bond can also be protonated, producing a secondary carbocation which is captured by $\ce{Cl-}$. If both happens, the product would be 2,4-dichloro-4,5-dimethylhexane.