So I was playing 5e a while back, and we were ambushed by a large group of kobolds. I had to run past them taking damage from attacks of opportunity. If I could get out of their reach with a high jump that would be for the best. I wouldn't have almost cost us the match. How can you tell how high a PC can jump under normal circumstances? I know how far they can jump, but it does not explain anything about height.
Moving one square vertically will trigger an opportunity attack.
The relevant general rule is simply:
You can make an opportunity attack when a hostile creature that you can see moves out of your reach.
(PHB, p. 195, "Opportunity Attacks").
And the rule for figuring ranges on a grid, which includes reach, is:
To determine the range on a grid between two things ... start counting squares from a square adjacent to one of them and stop counting in the square of the other one.
(PHB, p. 192, "Variant: Playing on a Grid")
D&D is an exceptions-based system, meaning that general rules apply unless there are more specific rules in place, and there's nothing in the single paragraph of rules for flying in combat (PHB p. 191) that change how these rules would operate.
So, if C1 can make melee attacks against C2, they are within one square of each other -- that is, they are adjacent on the grid. If C1 moves so they are more than one square away from C2, C2 can make an opportunity attack (and vice-versa). On a grid, all movement is in increments of one square, so any move from either combatant that puts more than one square of distance (making them no longer adjacent) between them is sufficient.
How big is a vertical square? The rules just say that a square "represents 5 feet". There are no specific rules for how to use a grid for three-dimensional combat (and remember that the rules for a 2D grid are themselves both optional and lightweight). The most direct interpretation of the rules is that a square on the grid represents a 5' cube of space. Thus while an active, flying, Medium creature will, strictly speaking, have bits of them sticking outside of the boundaries of "their" cube, what really matters is that any creatures who occupy any of the 26 adjacent cubes are within their one-square (or one-cube) reach.
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.