I recently retired from the Navy. I was stationed in Japan from March 2000 - May 2005.
Every ship has space to store things. Generally most ships make space to store a seabag of whatever they might need in the berthing areas. But, it can be difficult to retrieve it since the area is a big cage with bags tossed on bags. Additionally, every work center seems to have a place for storing items. My own work center had 3'x3' lockers in our work space. I would highly recommend that your child consult their fellow shipmates upon arriving, to clarify the storage capabilities available to them, before sending any Pelican cases, etc.
Qualifications are not that hard to complete, providing they are not anti-social. Those that meet people and socialize find that qualifications turn into who you know if you want to knock them out fast. I would not worry so much about finding time to complete them. They always get finished in time, since nobody likes to be singled out as delinquent. Most ships put new check-ins into an INDOC class that gets them qualified basic firefighting and maintenance before heading to their workcenters. The hard qualifications are done before they start working in their rates.
Concerning rental of gear. It is prohibitively expensive for junior enlisted to dive in Japan. The yen is currently 102 yen to $1. Which is not bad, until you factor in the high cost of living in Japan. The language barrier also greatly increases the difficulty in getting in dives, unless they hook up with other sailors thru MRW or some such group. The waters around Yokosuka are very cold and disgusting.
Deployments are the keys to diving when stationed overseas. But, I would encourage a first port visit to a country to be more about experiencing the local customs and seeing the sites. Some places are only visited once every 5 years, if visited at all. So, if they decide to return to dive, they know how to get around and about where they might like to revisit.
EDIT: Recommend they visit HIS (Japanese travel company) when booking a flight to another Asian country. They got me killer deals to Singapore, Thailand, Phillippines, Indonesia, etc. The Military travel company (SATO) can only book with US airlines, which limits the flight destinations and increases the cost by almost triple sometimes. Stationed in Yokosuka will make it easy to fly anywhere out of Tokyo, which is only an hour or two hour train ride from the base, depending upon the rail line you hope on.
