Although movement in 5e is not governed by squares, they are probably a helpful model to this particular exercise. Everything in the universe is effectively measured in 5' increments and using squares to model this will give us an effective answer to this question Let's posit a 50' wall
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Now let's look at the veritcal space next to the wall
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Finally we need to have the flat part on top of the wall, and the space above that
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Now, supposing our hero starts at the first y (marked a below), and runs up, he runs out of movement at the 10th y (marked b below).
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ax
If you start at a, you might be able to climb up onto the top of the wall. However, if you have a 45 foot wall you are safe. If you have to move to a, a dash would be required to not fall (as you don't have the ability to stay vertical).
The problem here is that if the wall extends at all into the square above the one marked b, then you're in trouble, you have to move into the square above it as diagonal movement is not possible.
This will be both at the discretion of the map makers (if the building extends a bit above the square you're probably not going to move diagonally) and also your DM (he may allow a bit of wiggle room here). Consult your DM before you attempt any 50' wall climbs.
45' wall climbs (and lower) are safe though.
Grappling (PHB, p. 195) states:
When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.
Movement and Position (PHB, p. 190) states:
Your movement can include jumping, climbing, and swimming. These different modes of movement can be combined with walking, or they can constitute your entire move.
(1) Therefore, if the steeder has a creature grappled, it won't be able to jump, unless the grappled creature is small or smaller. Alternatively, some effect might have increased the steeder's speed to 60+ without a grapple, and the halved speed is therefore still 30+.
Note that dashing doesn't help you here - it only increases your movement (PHB, p. 192), while your speed is still unchanged (e.g. at 15 feet).
(2) The steeder's leap is different from the regular jumping rules in the PHB, hence, all relevant rules can be found in the steeder's description. Since it can either expend all its movement to jump 90 feet or not at all, there is no halving of the jump distance.
Either it can jump 90 feet, or not at all (barring the regular jump rules, which of course still apply).
(3) Grappling also states:
The condition specifies the things that end it, and you can release the target whenever you like (no action required).
Breaking up your move (PHB, p. 190) only mentions
You can break up your movement on your turn, using some of your speed before and after your action. For example, if you have a speed of 30 feet, you can move 10 feet, take your action, and then move 20 feet.
Now, releasing a creature doesn't require an action, but I think the intent is clear - yes, you'd be able to release it during the jump. That also makes sense from a logical point of view, which is what I always try to take into consideration when I make any rulings as the DM.
Best Answer
No, it cannot.
You quoted the relevant text already:
The rules on dashing (PHB, p. 192) state:
When you leap for the first time, you have to use all of your currently available movement which, at this moment, is all the movement you have on your turn.
Now, when you take the Dash action, it creates an issue - you now have movement left, but this movement is still on the same turn as the leap. Hence, you now have retroactively invalidated the first use of the Leap, since that requires you to use all [your] movement on your turn.
How to deal with this issue isn't clear from the rules, so your DM will have to make a ruling. I'd personally rule that the first leap isn't affected and you can now use the movement you gained from the Dash for moving.
What you cannot do with this movement is leaping a second time. Assuming you / the Steeder has a speed of 30 feet, you go through the following process during your turn:
Note that all of this is just rules-as-written. If your DM decides that a Dash allows a second leap, that's not too far-fetched, from a logical point of view.
Bear in mind, though, that some features might allow the steeder to use Dash as a bonus action, hence allowing it to jump 3*90 feet, i.e. 270, in one turn.