I fully realize that this feat selection is going to be the most controversial part of this race, but I still feel like it's a great idea. Yes, only the human variant gets bonus feats, but what's to say there couldn't be a homebrew tiefling variant that got a bonus feat for some reasonable tradeoff? Feats IMO are in fact an option for races in general, if they're balanced. The feat is also limited to small list of them, so it's not even as beneficial as the human variant, making it a bit less of a balancing concern.
I think the feat helps the player get more excited about creating a bestial race like this since the the player can really solidify the character's raw bestial power through this special ability. For example, an archer centaur Fighter with Sharpshooter would really live up to the legendary marksmanship abilities of centaur, or a Barbarian centaur with Polearm Master, raging as they closed the distance, would be absolutely terrifying. My Druid player is actually going to pick Tough, so his centaur is going to have a huge sack of HP, which I think is also quite appropriate for a centaur of nature: it knows the magic, but this druid is built like a tank.
If this is for your homebrew race, that's all fine. My comments are all from the perspective of what would make a good general-purpose race for use at most tables. Bonus feats for races are not the approach 5E took because it's a little too complex and puts a little too much power into the race selection.
Balance-wise, you've chosen some of the very best feats. I translate "player gets more excited" as "we all know this is slightly overpowered." Toughness feat is +2 HP per level
on top of the racial Constitution bonus; the centaur
druid will basically have as many hit points as a mountain dwarf
barbarian.
Granted, the mountain dwarf is getting a ton of armor proficiencies, which would cost several feats on its own, and as you point out the variant human gets a bonus feat. So I think you're still in the neighborhood of "balanced" when considering the centaur's myriad drawbacks. And I'd still advocate removing the AC penalty, as the other penalties for size and shape are already bad.
I'm not familiar with the "normal squeezing rules" you're referring to, but if I don't know about it, I suspect other DMs and players might not know about it either. Even if there is slight redundancy with RAW, I think saying something about this challenge is expected of a centaur race description and it very well may be the case that DMs simply don't let centaurs through normal doors, justifiably so (and therefore could use this trait in their justification). Imagine trying to navigate a large building with normal doors to every room -- if the 9' Large centaur can just pass through them all without issue, it would seem a little broken. The centaur should think carefully about each door he/she goes through, and should expect an innkeeper to have a major "WTF?!" reaction.
PHB p.192. These are not obscure, secret rules buried in an appendix of the DMG.
That said, I think you are right that a "reminder" about common centaur scenarios is a good idea. I'd just advise that you make sure the rules you mention here line up with the official general-case rule.
You might want to mention the doorway problem in the entry for Large size. Most racial write-ups don't say much in their "Size" entry because there isn't much to say. But it seems like there is a lot for centaurs.
Normal rules for "centaur riding"? I do think it should be explained, and it's important to limit it to Medium or smaller. About the "embarrassment" details: it's almost cliché that centaurs do not like to be ridden. They shouldn't have the limitation removed, but it should always be roleplayed at least once: "You want to ride me?? I AM NOT YOUR HORSE!!" This is another instance where I want the race description to help explain what should be expected of a centaur.
PHB p.198.
As with the above, I think it's fine to remind people about the rules for common scenarios, as long as it follows the general rules. However since there's no actual mechanical impact here it might be better to put this in the flavor-text than the class traits.
I should also add that the MM centaurs are "monstrosities," not "fey". This is controversial, especially since official descriptions of the Feywild say that centaurs live there. It's a minor quibble but I think this reclassification should be made in worlds that feature centaurs and the Feywild prominently.
This reasoning makes sense to me.
I also find it weird that creatures like centaurs, harpies, doppelgängers and medusas are "monstrosities" but merfolk, thri-kreen, kuo-toa and lycanthropes are "humanoid" while off to the side we've got satyrs and dryads and pixies hanging out on the material plane being "fey." I guess I understand why they made it this way, but it seems confusing sometimes.
Another size-problem I just remembered: food and water consumption. It's probably worth mentioning that centaurs eat a lot more. And another size benefit: you can carry a lot more. Like a LOT more. I'd say both multipliers should be about 4x as much as a humanoid. Note that barding only weighs 2x regular armor, and the centaur's weapons are medium-sized, so the centaur PC should still have a lot of carrying capacity left over.
Also I think the listed weight slightly too much. Each doubling of creature size should increase weight by 8x. Based on human weight (114-270 lbs., by the book), this should put centaurs in the range of 900-2150 lbs. Googling "how much does a horse weigh" gives a range of 840-2200, which is pretty close to that. Now a centaur replaces the horse's head and neck (which is not very light) with a human upper torso. Let's call it half the weight of a human (57-135 lbs.) which gives us a final total of 897-2335 lbs. when added to Google's numbers. Rounding, I'd say 900-2300 lbs. for centaurs, which is significantly less than your initial specification.