Repositioning cruises tend to be in the spring. We have done a couple one on QE2 no longer available and Royal Caribbean Jewel of the Seas a lovely ship. Children are very rare on transatlantic trips. These trips offer excellent value for money and sea days are great for dressing up. Formal nights are on sea days. The officers also enjoy these trips very much. Recommend a crossing one way by sea and November is also good for getting out to Caribbean and flying home easy enough too. I would not go on any ship under 70,000 tons though as too much movement for my liking.
The caribbean hurricane season normally lasts until November and the repositioning cruises tend to go in the other direction(UK/Med to the caribbean) in the late autumn.
Caribbean to the Med/Uk cruises are most popular in the spring.The main problem with returning to the UK is the fact that the last 2 to 3 days of the cruise involves cruising from somewhere like Madiera through the Bay of Biscay which is notoriously unpleasant at that tme of year. Personally I would suggest finishing the cruise somewhere in the Med as a preferred option.
With regard to preferred cruise lines you pay your money and take a chance.The larger the ship the more stable they tend to be,but it all depends on your preferences.