We did the Mediterranean on the Millennium this past August...We are somewhat restricted to July and August due to school schedules--even when we don't take the kids, because my wife works for a school district...
August is not optimum, unless you really enjoy hot weather--May would really be best, but we can't all travel then...Personally, I have no problem with the heat--remember, Mediterannean climates have extremely DRY summers--so, although it's hot, there is little humidity, making it quite comfortable...just dress cool...
The seas are quite calm --we found them to be better than any of our Caribbean ventures (which always seem to be hurricane season)...no need for any concern...
We went with Celebrity due to a number of reasons and it turned out perfectly--we loved it...
One of the most important aspects of a Mediterranean cruise is the itinerary ...and the Celebrity itineraries have some major advantages...They do "one-way" routes--Venice to Barcelona or vice-versa...This allows you to do different pre- and post-cruise stays in the two excellent and different ports...Second, no "back-tracking", meaning more time, more time in ports and more ports...RCCL's Brilliance, for example, alternates routes between one thyat does Venice and the Adriatic and another that includes Athens and the Aegean...The Celebrity route took us to BOTH Venice (and Dubrovnik and Corfu in the Adriatic) AND Athens and Santorini in the Aegean...
The ship itself was magnificent and the service superb...
On our sailing, there appeared to be quite a large number of families with kids of various ages...including a lot of families from Europe...The kids all seemed to be enjoying themselves...I would not hesitate to do the Mediterranean on Celebrity with the kids...I don't think I'd do HAL with the kids...We took our kids with us on Princess in the Baltic a couple of years back...and the kids absolutely hated Princess and the ship...it was in August yet there were maybe only 20-30 kids out of 2000+ passengers and our kids were mad...nothing seemed to be geared toward them (To be truthful, everything, even the entertainment, seemed geared to the octogenarian crowd)...
We were so impressed with the Millie that we're taking the kids with us to Alaska this July on the Summit...
Have fun...
By the Way, you can read my entire detailed review here:
http://www.cruise-addicts.com/reviews/readreview.php?id=0000002441