Yes, the jump is the distance from the ground to the bottom of your feet

When you make a high jump, you leap into the air a number of feet equal to 3 + your Strength modifier [...] Either way, each foot you clear on the jump costs a foot of movement.

Intuitively, a vertical jumping distance makes the most sense as the distance between your shoes and the ground but really your entire body is moving the same distance upwards. The rules' language talks of the distance you "clear" and "leaping into the air" both of which evokes the imagery of the gap between ground and shoes. More direct evidence can be found in the following passage:

You can extend your arms half your height above yourself during the jump. Thus, you can reach above you a distance equal to the height of the jump plus 1 1/2 times your height.

So it is clear from the rules that when you jump, the distance is measured from the ground to your feet and that means that the rest of your body moved vertically that same distance.

If the PC can jump X feet, their feet reach X feet

If a 6' tall character jumps up 8' it means that their feet are 8' above the ground and their head is 6' + 8' = 14' above the ground. If they raise their arms the tips of their fingers would be 1.5 * 6' + 8' = 17' feet above the ground.

In this case, if a character jumps up at a 8' wall with a jump of 8', then the assumption is that they land on their feet on top of the wall.

The way the jump looks narratively is not defined

How all this jumping and leaping looks is completely situational and also not at all defined in the rules. In cases like these, the DM must fill in the blanks.

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.