If a team kicks off and the kickoff hits the upright of the field goal (or any part of the field goal) and the ball bounces straight back into the field of play (never hitting the ground out of bounds). Is the ball live? Can the kicking team recover it?
No. By NFL Rule 7, Section 2:
Article 1: Dead Ball Declared. An official shall declare the ball dead and the down ended when:
(h) when any legal or illegal kick touches the receivers' goal posts or crossbar, unless it scores a field goal
This includes field goals, kickoffs, and punts.
The NCAA first considers the goal posts to be out of bounds (Rule 1, Section 2, Article 5.a), then makes an exception for the case when a kick hits them and bounces through (Rule 4, Section 1):
ARTICLE 3. A live ball becomes dead and an official shall sound his whistle or declare it dead:
a. When it goes out of bounds other than a kick that scores a field goal after touching the uprights or crossbar [...]
This is not the case in arena football, though. Arena football has such a small field that it is necessary to consider the uprights and the surrounding netting as inbounds, or else no kick would ever be returnable. From the Arena Football League rules (short version):
The receiving team may field any kickoff or missed field goal that rebounds off the net.