All interesting questions, but not easily answered. Sea year experiences vary hugely from person to person, ship to ship, and crew change to crew change. It is generally a time where Midshipmen (called cadets while they are aboard ships during sea year) get a lot of hands on experience with their future career possibilities. Engine cadets report to the engine department, while deck cadets report to the deck department. Usually two cadets are assigned per ship, one deck and one engine, and they are referred to as sea partners. (The exception to this standard is on MSC ships which generally have more than two cadets.) You generally get to select your own sea partner as long as you are the same split and the opposite department. How much you get to do as a cadet varies depending on the company, your supervisor, and your performance.
Getting sea projects done can be difficult. It is expected that you will work a full eight hour or longer day, usually seven days per week, then do your projects on your own time. It can be done, but it requires self-discipline, particularly second sailing where it is easy to procrastinate when you know you have eight months to complete everything. Sometimes you get lucky and only work six days per week so you can work projects on your day off. Daily schedules vary a lot depending on what ship you are on, whether you are in port or at sea, and if in port, whether or not you get to go ashore.
Many different types of ships are available; yes, including tug boats and massive cargo ships, as long as they are U.S. flagged. You will learn more about the different kinds of ships available while at KP. There are advantages and disadvantages of each. You also may want to graduate with a particular kind of endorsement on your license which will require sailing on a particular class of ship and performing specific tasks.
Yes, you can go on Navy ships of many different types during part of your sea year. You can have opportunities to go to Antarctica, but they are limited and competitive to get, and usually only two cadets per split. It is more common to be able to go to Alaska or near the Arctic. Although you may have some input into the type of ship and region you sail in, you are assigned a ship by your ATR (Academy Training Representative), and you may need to accept graciously whatever ship you are assigned in order to get your required sea days in.
Regarding A or B split, there are fans of each. Most mids are assigned depending on what sport they are in so they do not miss the season their sport is in. If you are not in a sport you may get to choose. Which you prefer may depend on what split most of your academy friends are in, how you like New York summers or winters, how important it is to you and your family to have you home for Christmas, and lots of other considerations.
Keep in mind that you will have a specific class at KP before you go out to sea which will explain all your options and obligations during sea year.
I've done that more than a few times with APL on the PNX run.. Ya better hang on for dear life in the Gulf of Alaska in the winter.. but as an old Chief Mate I sailed with used to say 'Get used to it kid.. you're goin' to sea, not to Sunday school..' 