In the infield, a pop up has a natural tendency to move back towards the infield, happens down either line and in the area around the plate. So to compensate for the ball moving towards the infield, the catchers glove is positioned so it can move with the ball, rather than stabbing at it.
'cause it's an out
Yes, they do have to tag up, unless the fielder, for whatever reason, does not catch the ball.
Basically every fielder, both infielders and outfielders, have the same job: stop the ball. The infield is more about quick ground balls being stopped. The ones in the air are generally easier to stop and catch.
Field and get the ball
yes
If a batted ball hits the ground before both (1) leaving the infield and (2) a fielder has a chance of catching it, that is (generally) considered a ground ball. If it leaves the infield without touching the ground or a fielder has a chance of catching it, that is considered a fly ball. A ball that does not much of an arc to its motion is often called a "line drive" instead of a "fly ball."
The runner is also out and the ball is dead. All runners go back to previous bases.
Most of the time, yelling "I've got it!" or "Mine, mine!" is enough to tell the other fielders that you are going to catch the fly ball. However, if two or more fielders are calling that at the same time, it can be difficult to both to hear, and can result in a collision. However, some teams set up a "primary fielder," which means that, for example, for any ball hit between two fielders (center and right, or center and left), one of them (usually the center fielder) is always the primary fielder and the other fielder (left or right) will move to the back-up position. Only if the primary fielder indicates he or she cannot catch the ball (for example, by yelling, "No! No!") will the other fielder attempt to catch it. The same basic strategy can be used for the infield.
For there to be a legal catch, there must be a catch and voluntary release of the ball. Therefore, if the ball knocks the glove off the fielder's hand, he didn't meet either of the criteria for a catch: he didn't catch it, and he didn't release the ball voluntarily.
The only other way I can think of that a runner would score from a fly-out without tagging up would be when the fielder catches the fly ball and then throws it into the infield and the infielder doesn't catch the ball properly, allowing the runner to advance home on an error.
Any fielder is allowed to catch any fly ball he can. If you are referring to the infield fly rule: under this rule, the hitter is called out and play continues as if the fly were caught even if no one catches it. It does not matter who catches it, but if the ball drops, the runner can tag up and run at his risk. The rule is only called when the ball seems certain to be catchable. The purpose of the rule is to prevent an infielder from intentionally dropping pop-up in order to get a double play.
5 secs
This is up to the discretion of the official scorer. If the scorer feels that one of the infielders had a clear ability to catch the ball and failed to do so, it will be ruled an error. If the ball hits a fielder's glove, it will almost definitely be ruled an error.