Yes, only ports with naval base improvements will repair vessels. Do check naval spending too, as the lowest (30%) is not very effective.
Yes, you can build ships anywhere. Certain ships require certain port levels, and some require the base to be part of the landmass of your capital. The clippers and frigates can be built at any naval base.
Yes, your ships will eventually sink.
My tactic for Japan is to surround them with war ships. Then invade it at either the northern or southern islands or both at the same time, and then slowly devour their troops. Make sure to do this before they westernise. It will be a lot harder once they westernise.
Your final point, I have no idea. I only know of the straits of dover and hormuz allowing france/britain and Persia/Abu Dhabi respectively. Well..there's the one in Zanzibar too.