What kinds of Foul Balls can be caught for an Out? May runners tag up on such Outs?

All of them, except for when the ball touches a foreign object such as a batting cage or fence (13.02d). Other fouls, including kicks on or above the knee (13.02e), balls touched more than once by the kicker in the kicking box (13.02f), and balls kicked outside the kicking box (13.02g) MAY be caught for Outs, and runners may attempt to tag up and advance on them, provided the caught ball is not in the possession of the pitcher on the mound.