If the ancestry doesn't specify Large (tall) or Large (long), then there isn't a definitive answer. But I'd consider it reasonable to rule they only have a 5ft reach. I'd note that the Centaur in the bestiary doesn't have more than a 5ft reach, so that would seem to be consistent.
Regarding corridors, the rules aren't 100% clear. In another thread I listed some arguments for why I believe the default for many large-sized creatures would be to treat 5ft corridors as a "tight space" which would only make them difficult terrain rather than requiring a squeeze check. When it comes to actually fighting in a tight space, there don't seem to be any rules at all regarding this beyond saying it is difficult terrain.
Regarding Climb checks, RAW doesn't seem to have anything that'd penalize Centaurs - but then again the official paizo rules are primarily made for biped ancestries, so I'd say it's on the 3PP for not thinking to include anything about this. Even if you consider something similar like a Horse animal companion - RAW doesn't seem to address the subject. So overall, I'd say you basically have to adjudicate this yourself, and it is probably best to let the player know how you'll run the situation well before it ever comes up at the table. Personally, I'd probably give them a -2 to -4 circumstance penalty along the guidelines of the Special Circumstances section in the CRB.
Regarding riding another PC, that is actually mentioned in the Gamemaster Guide under Different Types of Mounts. Overall, it basically discourages the idea of PCs mounting another PC - but says if you want to allow it you should require one of them to use at least one hand to hold onto the other, and both should have to spend an action on each of their turns to remain mounted. For the case of a centaur, I'd say you could reasonably ignore the hand requirement since that isn't necessary for a normal horse, but the action requirement seems to be for balance reasons (I'd theme it as having to keep yourself aware of the other PC's actions to avoid falling/throwing them off - and having slightly less time in your turn as a result).
I'd also note that normal mounted combat rules have both creatures share a multiple attack penalty, and the GMG description of mounting a PC says nothing about changing this - so I'd view the exact interaction as somewhat unclear since the two PCs don't share a single turn. By default, I'd be tempted to rule that the PCs have to move their initiative together and essentially share their turn (including MAP) much like how standard mounted combat works (after all, a centaur using a weapon would get in the way of their rider trying to swing/aim a weapon from their back at the same time). That said, you could potentially rule that part of each PC spending an action each turn to remain mounted is to avoid interfering with each other and not require them to coordinate their attacks in regards to MAP (which the PCs would probably prefer, though I'm not sure what issues it could create).
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As for other things to look out for - I'd consider if there should be any penalties for attempting to fight in a tight space (as in 1st edition). I'd also consider how the ancestry should interact with doorways during combat initiative, which should probably require squeezing, but the Squeeze action is made for exploration mode and measures distance in 5-10 foot increments when squeezing through a doorway that is probably roughly 1/2 a foot of travel distance. So if the centaur needed to get through a doorway during combat (such as the standard kick down the door leading to entering combat), you'd have to make some type of custom squeeze action to allow the centaur to pass through the doorway without saying they spend the entire combat just trying to enter the room.
Splash and emanations will cover more squares.
Multiple people can qualify for flanking with them in many cases and it becomes easier to qualify as a flanker for multiple allies.
It becomes very hard to stealth, take cover and similar.
You need to make equipment of their size available.
They will be immune to grapples, shoves and the like for small sized creatures, and will be able to grapple gargantuan creatures with titan wrestler.
There will be inevitable and endless discussions about centaur biology at table that may sidetrack the game a bit.
One of the big problems however, is this line under the rules for worn items;
"It’s assumed that items are meant to be worn by humanoids; any item that can or must be worn by a different type of creature either states this in its description or has the companion trait"
Centaurs are beasts instead of humanoids, so RAW they can't use most worn items (unless the authors wrote a specific ability or trait that lets the ancestry get around this) - this might make sense for magic boots, but seems pretty weird when it comes to say, a magic hat or magic gloves.