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.

It does not grant extra benefits

The grappling rules do not say anything about size other than that you cannot grapple something two or more sizes larger than you, and that your speed is halved during a grapple unless the grappled creature is two or more sizes smaller than you.

By RAW, no advantage due to size is granted for any grapple checks.

The Enlarge/Reduce spell has multiple effects. One is to enlarge or reduce the size of the creature. Another is to grant advantage on Strength checks and saves. That is, the increase in size does not cause the advantage to Strength checks and saves. Both effects are caused by the magic of the spell.

In the spell description, notice how the advantage benefit is written. It is said as "also, you have this benefit" and not "because of that, you have this benefit" -- suggesting non-causally related items in a list.

Enlarge Reduce

...

Until the spell ends, the target also has advantage on Strength checks and Strength saving throws.

If it was the size change that conferred the benefits of advantage, then a creature who was Medium size and enlarged to Large size, when grappling a creature of Large size, should not get advantage on their checks. But they do, because magic, literally.

Bigger monsters typically have higher Strength scores

In a way, being bigger does have advantages for the sake of grappling. But this is reflected in their high Strength scores and proficiency in Athletics check (if they have it). It is not reflected by any application of the advantage/disadvantage mechanic for reasons owing purely to size.