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.

No. The Amazing Light Speed Horse™ cannot move an infinite distance

(And what distance it does move without taking the dash action is at a speed of at most movement per round, and with the dash action is 2×movement per round.)

Per the definition of movement from the Basic Rules v 0.3, page 63, and also from the Player's Handbook, page 181:

Every character and monster has a speed, which is the distance in feet that the character or monster can walk in 1 round.

(Note that the riding horse is stated as a monster in the Player's Handbook on page 310, and the warhorse on page 311.)

Your exploit only works if you incorrectly substitute turn for round under an assumption along the lines of somehow the horse receives one turn for each humanoid rider's turn, rather than one turn per round. While the second individual could attempt to mount the horse on their turn in your scenario, the horse will already have used all its movement for the round.