[RPG] Can a monster with a flying speed lift a Grappled PC and then drop them

Grappling doesn't care about weight, only size matters.

The grappling rules do not distinguish from grappling a 300lb Medium-sized creature from a 100lb medium creature. If you're grappling a medium-sized princess, it doesn't matter how heavy she is, your speed is always halved, unless the creature is two or more sizes smaller than you.

So, in your example, the Barbarian could only move 25 feet then dash for another 25 feet. And, yes, they can do 5d6 damage, if everything lines up for them. On subsequent rounds, the Barbarian could just move back down, grapple again and dash up 25 feet. Equating to 2d6 damage on subsequent rounds, not that impressive any more.

It should also be noted that carrying/lifting/dragging an object is governed by different rules from carrying (i.e. grappling) a conscious kicking screaming princess.

No.

You cannot take a 5-foot step upwards because doing so would take at least 10 feet of movement, which invalidates the 5-foot-step action and becomes a Move Action instead.

Anytime you cannot spend only 5-foot of your movement to enter a square, it should require a move action, and thus it is no longer a 5-foot step.

You can move 5 feet in any round when you don’t perform any other kind of movement. Taking this 5-foot step never provokes an attack of opportunity. You can’t take more than one 5-foot step in a round, and you can’t take a 5-foot step in the same round that you move any distance.

If you moved 10-foot, it is no longer a 5-foot step. This has been discussed on paizo's messageboards here and here.

According to James Jacobs (Pathfinder's Creative Director), a creature should not be able to fly upwards as a 5-foot step.

The only exception for this, back in 3.5, was creatures that had Perfect maneuvering, which allowed them to move upwards without using double their movement, so they could move 5 feet upwards without using 10 feet of their movement. This exception no longer exists in Pathfinder, but if a creature has a similar ability, the exception should apply.