RAW
It appears to be two actions, honestly much to my surprise.
Move Action
You can take a move action in place of a standard action.
So when you do that, you get two move actions. You can use both to move up to your speed, as separate actions.
Nowhere under full-round actions is any mention made of a double-move action. Running and charging, yes, but not the simple double-move.
Now, Swarm Attack reads as follows:
they deal automatic damage to any creature whose space they occupy at the end of their move (no attack roll needed).
I read “their move” here to be “their move action” rather than “their movement.” I do this because, grammatically, “move” is not really short for “movement,” while having an implicit noun for an adjective (i.e. action for adjective move) is quite common in English.
So yes, this would trigger Swarm Attack twice.
However, it’s always good to acknowledge ambiguities in the rules.
Should it be run this way?
I’m not sure. I’ve never seen it run that way, and I sort of doubt that most swarms were designed with that ability in mind, but I could be wrong; after all, I was wrong about this working in the first place. In any event, just because the authors intended it one way doesn’t mean that’s how I want to run it in my game.
For my games, I’d have to carefully consider the swarms I was or was not using, and what options my players had for dealing with them, in order to determine how I would rule this. In my experience, swarm damage tends to be low, but at the same time the majority of character classes have almost-zero ability to do anything to them.
So, for example, if my party consisted primarily of mundane classes, I really would not make swarms twice as dangerous as I thought they were, because I already considered them too dangerous by virtue of their effective invincibility. So, and I suppose it’s a houserule considering RAW, I would not allow swarms to double their swarm attack damage by double-moving.
On the other hand, if I was making a campaign centered around swarms, and expected my players to build specifically with swarms in mind, I probably would go ahead and rule this as RAW, and let swarms do that. This would allow the swarms to be more of a threat for situations where the players aren’t just enduring them, but are actively and effectively killing them. It would also open up a few more tactics for dealing with them, such as partial action denial, à la Nauseated.
If I anticipated swarms being an issue either way, I’d make a point of informing my players of this issue, since it’s unlikely that any of them have considered it on their own.
As for “verisimilitude,” or “seeing it through the game world’s lens” or whatever, that doesn’t really come into play here: swarms are explicitly an abstraction, particularly with respect to the Swarm Attack (using one lump sum rather than thousands of minuscule attacks), so it’s really impossible to say what makes more sense in-character; in-character, a completely different thing is happening.
Your friend is basically wrong.
The mounted combat rules are not very well written. But the crux of the matter is that there is a distinction between you charging while mounted, and your mount charging while you are mounted.
From the Mounted Combat section of the SRD:
Your mount acts on your initiative count as you direct it. You move at its speed, but the mount uses its action to move.
Mechanically, the mount spends its action to grant you its movement, rather than moving directly itself.
In other words, when you "move" while mounted, the mount spends actions. When you charge while mounted, the mount spends actions, but you are still the one who is charging.
Particularly bad is this paragraph:
If your mount charges, you also take the AC penalty associated with a charge. If you make an attack at the end of the charge, you receive the bonus gained from the charge. When charging on horseback, you deal double damage with a lance (see Charge).
You really have two different clauses here. The first is what happens when your mount charges (instead of granting you movement):
If your mount charges, you also take the AC penalty associated with a charge. If you make an attack at the end of the charge, you receive the bonus gained from the charge.
The second is the special rule for lances, which triggers "when you make a charge while mounted:"
When charging on horseback, you deal double damage with a lance (see Charge).
When trying to run at someone and hit them with a lance, the mechanics are:
You take the charge full-round action.
The mount spends its actions to grant you its movement.
You stop at the edge of your reach, and poke it with your lance.
By contrast, I suppose you could command your mount to charge.
The mount takes the full-round charge action.
The mount moves to within its reach of the target.
The mount attacks.
You may attack, if able (but generally not with a lance, because your mount is too close to the target).
Best Answer
As always, mounted combat is a complete mess.¹
In this case, though, it seems to me fairly clear² that, like many other “combination” actions, a double move exists in a state of superposition³ until you need to nail down exactly what’s going on. Compare full attacks, where you are free to decide you were taking an attack action or a full-attack action after seeing the results of your first attack. In the same way, you can decide whether your mount is taking a single move action or a double move action after the mount has completed its first move action—because until that point, it doesn’t matter if it’s a single move or a double-move (equivalently: you haven’t made an observation that determines its state one way or the other.) But at that point, you have to decide: whether it’s single or double affects whether you get a penalty on your attack(s), and whether you can continue moving. If you decide it is taking a double-move action, it can continue moving—and conveniently, that’s also the halfway point,⁴ so that same place is where you make your ranged attack⁵ from, if you elect to use that option.
Note that I’m making no determination as to whether the double-move is one action or two. This result stands either way.
Also not addressed in this answer: the Pathfinder FAQ that claims that the rider has to use the same actions as the mount every time the mount does anything. If that’s in force, mounted combat basically straight-up doesn’t work, so I recommend ignoring it with prejudice.
I’m not even restricting this statement to Pathfinder, mounted combat is pretty much always a mess in just about all “crunchy” systems.
Read: not actually clear at all, in fact we could definitely make arguments for other interpretations, but this feels most consistent and best able to cover all the rules without creating an awful mess to adjudicate.
Simultaneously both states and neither, in the manner of Schrödinger’s cat.
Technically, it’s where you have used up half of your potential movement; you could choose to move less than your total movement on either half of the movement and I feel it would still be a legal double-move, and the ranged attacks should still come from wherever you were when you chose to use another of your mount’s move actions to make this a double-move. Trying to use half of your actual movement is an exercise in misery since you would have to figure out the entire path, adjudicate anything that occurred while moving (attacks of opportunity, traps, etc.) that might affect how far you get to actually move, and then back-calculate the halfway point and retroactively make attacks—that’s a mess no one needs.
As I’m reading it, either a standard-action attack or a full-round action full-attack, at your choice. The wording could definitely be clearer on that, though.