Per the rules for jumping in the Adventuring chapter (Basic Rules p64, probably the same section you cite):

each foot you clear on the jump costs a foot of movement.

Yes, you can jump in combat as part of your movement; no, you can't jump further than you could move anyway. Rules as written, the Jump spell doesn't extend this distance.

If I were the GM, I would say that the Jump spell makes every three feet you clear cost a single foot of movement, but that would be a house ruling.

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:

1. It's at DM discretion (what isn't)
2. 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!