6-7 days is really pushing it from Melbourne unless your boat is a racer that's fast upwind. My last trip from Mooloolaba took 21 days including a 2 day stop at LHI so in an average cruising yacht I'd allow more like 14 days minimum, and prepare to be at sea for a week longer than that. Added to that, storms in the Bass Strait are just as likely to keep you in port for lengthy periods after your planned departure date.
There is a ridge of sea-mounts in a straight line between Sydney and LHI, almost exactly half way. They are probably just a bit north of a rhumb line between Melbourne and Auckland and about 1/4 of the way there. Yes, you should avoid those and the ridge, stay at least 10nm away because it can induce lumpy seas. On the trip to LHI we tried to make north of it while we could, because the prevailing winds are NE rather than E and we were fortunate enough to leave on a SW, but you may not be able to achieve that from Melbourne. Sea state can be bad in that area even in calm seas and gentle winds because of the convergence of the currents.