Yufuin and 7-day JR green pass

Does 7-day JR green pass cover the train from Hakata to Yufuin and vice versa?
(e.g. Yufuin no Mori, shinkansen from Hakata to Kokura, sonic, Sakura, local train etc)

I would like to take Yufuin no mori from Hakata to Kokura and then change other faster train to Yufuin, to shorten the time from 4 hours (all by Yufuin no mori), to 3 hours. then take the trains with the shortest time from Yufuin to Hakata in my return trip. Is it possible?

I know that Yufuin no mori has no green cars seat, but only ordinary reserved seat. But for shinkansen from Hakata to Kokura, sonic, Sakura, local train, do they have green cars?

If I buy 7 day JR green pass for my kid who occupy a seat, will there be any case that he need to pay addition charge, when he rides on any special trains? because I read from the Kyushu rail website "How to determine Adult and Child for train fares" that "Green Car tickets prices are the same as for an adult." for child Between 6 and 11 years old. but when I buy the 7 day JR green pass, it is just half that of the adult price. Pls clarify.