From the Player's Basic Rules, page 71 (or PHB p. 191):
In contrast, you can move through a hostile creature's space only if the creature is two sizes larger or two sizes smaller than you.
An ogre is a large creature and elves are medium. So an ogre can't move through an elf's space. In your previous example, if we have a 10-foot wide passageway with two elves standing side by side, the ogre can't get through without killing one of the elves or forcing them to move.
The rules on squeezing into a smaller space are for an ogre trying to move along a 5 foot corridor. The rules on creature size that you've quoted back this up—the ogre isn't actually 10 feet wide, that's just the space he controls. So he can move through a 5 foot wide gap, but it's cramped and he can't move freely.
Now, you might be thinking that it's a bit unfair on the ogre if the 2 elves can form an impenetrable barrier against it. As you've said, he'd rather shove them aside than squeeze between them. And he can do just that! From the Player's Basic Rules, page 74:
Using the Attack action, you can make a special melee attack to shove a creature, either to knock it prone or push it away from you. If you're able to make multiple attacks with the Attack action, this attack replaces one of them.
The target of your shove must be no more than one size larger than you, and it must be within your reach. You must make a Strength(Athletics) check contested by the target's Strength(Athletics) or Dexterity(Acrobatics) check (the target chooses the ability to use). If you win the contest, you either knock the target prone or push it 5 feet away from you.
So you're right, shoving is the ogre's answer here.
Now, with your new example, (2 elves in a 15 foot corridor with a 5 foot gape between them), things are different. The ogre does indeed have to squeeze between them. Fortunately for him, it's not going to matter all that much.
Why? Because he's almost guaranteed to be moving on his own turn. Unless there are more enemies than just the elves, he's not going to provoke any opportunity attacks while squeezing, and he can attack before or after he squeezes. So the only squeezing penalty that is actually going to apply to him in this situation is the double cost for movement.
Note that if he stopped between the elves, all these penalties would apply to him. That makes sense though—standing between two enemies with not enough space to move around in would make it difficult to dodge attacks or attack effectively.
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.
Best Answer
It May Depend on the Creature (and the DM)
The rules are unclear on how high a "creature's space" is, and a DM may have to make a ruling. Unfortunately, "a creature's space" is defined only in a 2-Dimensional sense: on PHB, p. 191, it is defined entirely in terms of squares (not cubes).
A DM will have to decide how tall a creature's "space" is when it comes up. This may be based not only on a creature's height (which might vary from about 4 feet for dwarves to about 8 feet for hobgoblins), but also by the space they control. As you stated:
So precisely how tall a "creature's space" is may be different than its height. Different DMs may decide this differently (declaring a 5 foot cube to be every medium creature's "space" or determining space based on height, or however they wish). In some games, the top 5 foot cube of the door might be considered outside of an enemy creature's space (although definitely within their reach, and thus subject to opportunity attacks).
In your campaign, it looks like the DM considers the top part of the door to be within the hostile creature's space. Thus, unless they are using the optional "Tumble" rules (DMG, p. 272, which allows you to make a contested Acrobatics [Dex] check as an action or bonus action to move through an enemy's space), you cannot pass through it.