"Do the mids get the whole week off between the end of plebe summer and the start of classes?"
"Week off" -- well, it's just not that kind of place... Plebe summer and summer training for the rest of the mids end with Reform of the Brigade, when everyone is due back if they are not already there for some reason. In the days prior to the start of the ac year calendar, midshipmen are busy moving from summer rooms to company rooms, getting mandatory briefings, getting class schedules finalized, getting books. Plebes are getting computers issued, books issued, more gear issued, and being introduced to the plebe training program in their company (read: more stuff to memorize and special plebe duties). Plebe training staff transitions the plebes into their ac year company. It's a beehive with no play time. Plebes especially. The returned Brigade is eager to ensure the plebes get properly introduced to their place in life.
Plebes will have some off-Yard liberty ("town liberty") on Plebe Parents Weekend. Depending on the briefing schedules, the duty roster, and this year's leadership philosophy, plebes may not have Saturday liberty for a weekend or two after Plebe Parents weekend. Once Plebe Parents Weekend is over, I would say most parents leave.
Plebes usually do not have "Yard liberty," as in, visiting with family on the Yard time, during the week. The Mid Regs and how their particular company enforces them will be explained very clearly to them. Plebes are also much restricted on use of the Drydock snack bar, as to when they can use it and even if they can sit down.
If the usual pattern holds true, spend quality time with your plebe on Plebe Parents weekend, hug him/her goodbye for a while, unless you live close enough to visit on weekends. It will probably be Thanksgiving before the next batch of quality time.
Let your plebe be the guide for visiting.