Answer

At the beginning, it is speaking about a Sabbath that is not Rosh Hashanah. There it is discussing practice in learning to blow. That, they did not permit an adult to involve himself with, because that would be directly feeding a prohibition into the child’s hands, which is prohibited by Torah law, and there is no law of education here because we are dealing with a minor who has not yet reached the age of education. But they did permit the children themselves, meaning there is no obligation to stop them, because as a matter of Jewish law, if a minor is eating non-kosher food, the religious court is not commanded to separate him from it.
The question is why in the latter clause, which deals with the Festival of Rosh Hashanah itself, they did permit it. It seems that he understands that this is not considered directly causing a prohibition. Therefore, if the child has reached the age of education, it is even a commandment in this matter, and if he has not yet reached the age of education there is still no prohibition. Why is this not considered directly causing a prohibition? Apparently because the blowing itself is a rabbinic restriction, and on the Festival of Rosh Hashanah they did not decree that rabbinic restriction, since in any case people blow on that day. In other words, blowing on the Festival is not merely overridden; it is permitted. And of course there is also no prohibition of muktzeh in this on the Festival of Rosh Hashanah.
Question 3 is not really a question regardless of this. If you are speaking about someone who has reached the age of education, then you are speaking under the law of education and not under the law of directly feeding a prohibition into a child’s hands, because that applies even to very small children. But under the law of education there is also an obligation to educate them and teach them to blow, and that overrides the educational requirement to observe a rabbinic restriction. But according to my approach, there is no need for that, because on the Festival itself there is no rabbinic restriction here at all.
The last approach I suggested is apparently the approach of the Raavad there, who writes that they permitted it even on the Sabbath adjacent to the Festival, and there there is certainly a rabbinic restriction:

Rather, this is the way to understand the baraita: in the case of a minor who has reached the age of education in commandments, his father occupies himself with him so that he may learn, even on the Sabbath adjacent to the Festival, as they said (Sukkah 42), “If a minor knows how to wave, his father acquires a lulav for him.” But a minor who has not reached the age of education—they themselves may blow on the Festival, but not on the Sabbath; and all the more so, his father does not occupy himself with him.