This is a fun looking but messily written archetype that trades the cavalier's traditional mount and most class abilities for a phantom-like mount and a couple of nice abilities. Due to the way it's written, it leaves a lot of questions about how the new mount works.
1) What feats can it take?
It's a phantom, not an animal companion anymore. RAW, it looks like it's thus open to any feat for which it meets the requirements.
2)Is the animal selected for the ghost mount anything but cosmetic and does size affect its stats?
The stats for a ghost mount is included on the page already and even for a halfling that picks a wolf it has a bite and two hoof attacks (...yeah) and lacks the typical trip. RAW, no matter what animal it looks like, it's a medium or large horse. And since the stats are the same for a large ghost mount does that mean it takes the -1 to all attacks and AC without gaining larger damage dice or reach? (And if a ghost mount horse has reach unlike a normal horse, then the wolf with hoof attacks suddenly is only the second strangest thing here.)
Additionally, phantoms don't change in size like an animal companion so the wolf would remain medium even up to level twenty. Although this makes sense since the wolf is technically no longer alive.
3)Does the increased slam damage dice that a phantom normally gets applied to a ghost mount's natural attacks?
This seems like a major oversight due to RAW wording since ghost mounts already loose a significant offensive ability due to no emotional focuses and since they can't deliver spells like a phantom. RAW the only way to boost a ghost mounts damage output is to use the cavalier's neck slot to wear an amulet of mighty fist. This is especially important since as a phantom they don't get iterative attacks.
3) Skills per level?
Do they get the normal 2 per HD like a normal phantom despite a lower intelligence? I tried comparing this to the typical phantom rules, but it's as big a mess as the ghost rider wording. Not counting their automatic bonus increase in the emotion tied skills, RAW they get two per HD but RAW if their intelligence INCREASES by only one or two, the number of skill points DECREASES to one per level instead of two. As is, I'd think the ghost mount gets two per HD, and no bonus from emotion, until this mess gets cleaned up.
4)Stacking with spiritualist levels?
This is a bit of a left-field question since they probably don't expect a ghost rider to multiclass into Spiritualist. Neither Spiritualist class description nor this archetype says the levels stack when calculating the phantom's level with other classes that grant a phantom. Or does this mean that a Ghost Rider/Spiritualist would have two phantoms, albeit only one manifesting at a time? Judging from PFS rules in this FAQ on cavalier mount and druid companion animals stack so this may stack since both are phantoms. However, since the ghost mount type phantom is unavailable to the Spiritualist, the levels may not stack akin to cavalier levels not stacking for druid bird companion.
I agree on the feats, it says "spectral companion functions like a spiritualist’s phantom companion" and phantoms "can select any feat it qualifies for, but it must possess the appropriate appendage in order to use some feats." Since there is nothing else specific about feats in the archetype (which has exceptions to normal phantom rules) that is what you get.
I believe the animal selected is purely cosmetic, although it implies that movement modes would have an effect (mentioning swim speed specifically) but not providing any mechanics for that. Poorly written indeed. The phantoms rules say that they are modified for size, but doesn't really explain how, other than different damage dice in the phantom base form, so I believe you are correct as far as larger ones getting nothing in exchange, since the mount form doesn't have that. What they really should have done is provide a few possible 'base phantom mount' stat blocks, rather than just one.
I agree that damage is a mess. That said, I think the point of this archetype is to have a mount that you can get back even if it dies, not so much a combat pet on its own, but a reliable creature for you to deliver charge attacks while riding. Apparently a phantom mount still gets the scaling damage, since the archetype doesn't say they don't, but it also doesn't explain at all how it would work with the mount base form.
That said, I think you are not quite right about the emotion. "Ghost mounts do not gain emotional focus abilities." It doesn't say they don't have an emotional focus, just they don't get the special abilities. So an Anger focused mount would get intimidate and survival skill ranks, good fort and will saves, bonus to str, minus to dex and increasing str instead of dex but they wouldn't get Powerful Strike, Aura of Fury, Ferocious Mein or Furious Wail. That is how I read it anyway. Still not great, but perhaps not quite as bad.
I agree phantom skills are a mess. I'd treat it as always minimum of 2 per level, but no more unless you get an int of 12.
For the Spiritualist levels stacking, I would expect that it wouldn't if you were playing PFS. Frankly, the rules are enough of a mess that even trying it would be a bad idea in that sort of environment, at least unless some clarification came out. In a home game, I could see a GM allowing it, along with coming up with descent rules to fix the issues you have highlighted.
Since the phrase "takes the form of" is used to describe the shape taken by polymorphed creatures and the shape taken by the ghost rider's phantom, I think it's fairly safe to assume that you treat your mount as a phantom with the physical characteristics they would inherit if under a polymorph effect changing them into an acceptable creature type. This would mean that they would likely need to be four legged in order to carry a rider.
So they should gain the creature's base speed, their natural attacks, and apply the correct size modifier while not adjusting their stats. They would also likely lose their slam attacks.
That said, an official clarification would be as welcome as it is unlikely.