yes, Ship upgrades carry through, and the Mega weapons are indeed only available if the original weapon was upgraded. (The weapons only have to be upgraded by XP, eg lancer becoming the heavy lancer, it doesn't matter if you don't have all the platinum bolt upgrades for the weapons) And don't worry if you haven't finished upgrading the weapon, the XP for it will carry over too, and since the enemies will be slightly stronger, those weapons should continue to gain XP faster than before.
Hope what I've said makes sense! 