I think that it is meant to take months to train a ship up to 100% That has been my experience at any rate.
As far as I know, you do not need to actively train fighters - just assign them to a carrier and they will 'train on simulators' whilst docked.
The carrier DOES have to be in a fleet that is undergoing training though. I often have a number of carriers in a training sqdn that are at 100%, because they are training up their fighter complement due to combat losses or upgraded fighter classes. I also have fighter bases around major planets that do nothing BUT train. It does present a minor problem in that the bases fall out of orbit of their planet as soon as you start this as the autoorders generated by the training routine probably shouldnt apply to these types of unit. I solve it simply by using the SM option to relocate the fleet whenever I need to do something to it. This also brings up a minor exploit - put a speed 1 unit into a training fleet then the whole fleet continues to train but doesnt expend fuel.