As to when they would learn to write, I'm not sure, I would imagine it would be at the same time you did, in early childhood.
As to why, the Voyager episode you mentioned proves that there are circumstances where a hard copy of information is preferable to a digital copy. Off the top of my head, I can think of another circumstance...
In a predominately digital/technological society, which The Federation certainly is, the safest way to make absolutely sure that there is no digital trace of a piece of information would be to never put it in digital form. Want to make sure some clever Star Fleet officer can't recover your deleted file? Never create it in the first place. Just write it on paper and make sure to destroy it. The intelligence/clandestine applications are easy to imagine.
In the specific case you bring up, I guess you could make an argument that Chakotay could have typed/spoken it and then printed a hard copy. I dont really have an answer there. Maybe he just prefers writing? Some people do.
However, that episode is not the only example of hand writing in star trek.
For example, in the DS9 episode "The Visitor" Jake Sisko is shown using a stylus to hand write onto a tablet. Later in the same episode, he presents a hand written paper copy of a novel to the girl visiting him. He says that it's full of notes and edits.
I suspect that, in reality, you are correct that there would be less emphasis on hand writing in that society, just like many schools don't teach cursive today since long form handwriting isn't used as much. But, just like we still need or prefer to write some things out by hand, I figure the same applies to Star Trek.