Rules
From the Player's Handbook:
High Jump. When you make a high jump, you leap into the air a number
of feet equal to 3 + your Strength modifier if you move at least 10
feet on foot immediately before the jump. When you make a standing
high jump, you can jump only half that distance. Either way, each foot
you clear on the jump costs a foot of movement. In some circumstances,
your DM might allow you to make a Strength (Athletics) check to jump
higher than you normally can.
p. 182
Falling
A fall from a great height is one of the most common hazards facing an
adventurer.
At the end of a fall, a creature takes 1d6 bludgeoning damage for
every 10 feet it fell, to a maximum of 20d6. The creature lands prone,
unless it avoids taking damage from the fall.
p. 183
Step of the Wind
... Your jump distance is doubled for the turn
p. 78
Jump
... the creature's jump distance is tripled ...
p. 254
Combining Magical Effects
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don't combine, however. Instead, the most potent effect - such as the highest bonus - from those castings applies while their durations overlap.
p. 205
This last gives several interesting possibilities for how these combinations actually work - but that is another question.
Definitions
Giving "jump" and "fall" their common English meanings (1.b & 2.b being the most appropriate), they are not the same thing. Jumping is voluntary and in control, falling is involuntary and out of control. So the first does not automatically lead to the second to my mind.
Designer Commentary
In addition there is this quote:
@Plaguescarred Can you jump farther than your movement when using
magic i.e spell Jump & boots of striding and springing?
@mikemearls
i'd rule yes - design intent is to make you jump super far
Read more: http://www.enworld.org/forum/showthread.php?362742-Jump/page2#ixzz3SirpzT8U
So we can draw two things from this:
- It's at DM discretion (what isn't)
- There's magic involved.
Discussion
Naked Jumping
Starting from the basics, a PC can jump up 3 + Strength modifier, without magic this is a maximum of 8' - so there are no falling considerations. A monster or magiced-up PC with a Strength of 24+ or a thief with Second Storey Work and high enough Strength and Dexterity can jump into the danger zone (>10'). I would argue that if you are strong enough to jump that high, you have the muscle strength to absorb the shock of landing without damage. There is some simple physics involved here - the speed at which you leave the ground equals the speed that you return - if your leg muscles can generate that speed then I say they can absorb it.
As a side note, I would rule that you can safely jump down less than 10' or equal to your maximum high jump whichever is the greater. Oh, and they wouldn't need a run to jump "straight" down. If you are jumping across a chasm to a lower ledge, I would rule that the so long as the total drop from high point to low point did not exceed this falling would not be a consideration.
Step of the Wind
A monk with Step of the Wind can double these heights. A 5th level monk wouldn't take damage from a fall like this anyway. For lower levels, as this is a specific class feature with a cost; I would be disinclined to nerf it.
Jump Spell
Jump can be cast on anyone and takes the maximum of a very strong PC to 24' and monsters to 36'. I cannot accept that this 1st level spell is intended to give the PC the power to jump up in the air (to whatever height) and then come down by falling to their death. This is simply not credible, particularly when you take into account the designer's comment on how he sees the spell being used.
The Combination
Wow! What a great idea the PC had! Of course you don't take falling damage and have some inspiration while you're at it!
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.
Best Answer
Under normal circumstances your high jump would reach 7 ft (3 + your Strength mod of +4).
Certain abilities and spells can increase this, so you would need to give more details on your monk.
Refer to this question
You seem to be under a bit of a misconception on what the jumping rule is actually saying and adding everything up(?). The high jump rule is saying that
If you have a run-up of 10ft then your height when jumping is 3 + your Strength mod. This is the number of feet you leap into the air.
If you do not have a run-up and just try and leap straight up then you only leap half that amount. i.e. 3 + your Strength mod, then divide that by two. In your case that would be 3.5 ft
The distance you jump is subtracted from the number of feet you can move in a round in order to calculate how much movement you have left. So if you run 10ft then jump forward 7ft, you would still have 33ft of movement left (50ft movement minus the 10ft minus the 7ft jump).