PAO by the RAW allows you to become an invisible stalker. While it doesn't specifically state you gain any abilities of the form you assume, it references greater polymorph, which in turn references the various "form of" spells. Which do explicitly state what abilities you gain based on what form you take.
So, by RAW, it's possible to gain any of the following abilities via POA
darkvision, whirlwind, earth glide, burn, vortex, flight, breath weapons, damage reduction, regeneration, trample, low-light vision, constrict, grab, poison, blindsense, scent, tremorsense, ferocity, jet, pounce, rake, rend, roar, spikes, trip, web, and probably others I've missed.
Invisible stalkers are elementals but their natural invisibility is not an ability covered by any of the elemental form spells. A DM could rule that you gain this ability if they feel it is on par with the other abilities that are possible with this spell. It's worth noting that even the fey form spells do not convey invisibility even though you could use them to look like a pixie.
Invisible stalkers absolutely have a physical form, they are not incorporeal and while they can negate invisibility purge they have no way to stop see invisibility.
All this to say. by the RAW, if you use POA on a creature and turn them into an invisible stalker, they would look like a non-invisible invisible stalker, which is a type of air elemental.
PAO is the most misunderstood spell in the game (and with good reason. It's not well written). Here is what it can actually do:
It functions as the spell Greater Polymorph (which itself functions as either: animal or magical beast as Beast Shape IV, elemental as Elemental Body III, humanoid as Alter Self, plant as Plant Shape II, dragon as Form of the Dragon I; Note that this means PAO cannot do aberrations, constructs, fey, monstrous humanoid, oozes, outsiders other than elementals, undead, or vermin), with the following exceptions:
1)The duration works differently, as shown in the tables (potentially being permanent)
2)It can turn creatures into objects, using the rules listed in the spell
3)It can turn objects into creatures (while it says it can do this, the mechanics for doing so are not fully explained, due to how polymorph changed between 3.5 and PF. For instance, what is the objects HD? BAB? base saves? etc.)
4)Finally, PAO can be used to mimic one of the following spells instead of the above uses: baleful polymorph, greater polymorph, flesh to stone, stone to flesh, transmute mud to rock, transmute metal to wood, or transmute rock to mud
Note that when polymorphing into a creature, you do not gain that creature's stats or special abilities. Instead, you gain bonuses to your own ability scores as appropriate for the polymorph spell (Beast Shape IV for animals, etc.), and any special abilities that polymorph spell lists, as well as all natural weapons.
In regards to this thread, you cannot PAO into an Invisible Stalker, because Elemental Body III can only turn you into a Large or smaller air, earth, water, or fire elemental (as in the creatures with the name "X elemental", not just any outsider with the elemental and air/earth/water/fire subtypes), which is what Greater Polymorph functions as, which is what PAO functions as
After closer reading I agree with Zollqir. The following line
You can use this spell to transform all manner of objects and creatures into new forms- you aren’t limited to transforming a living creature into another living form.
sounded like it allows you to turn anything into anything. But looking at it closer now, I see that it allows you to turn anything (except magical objects and things that are immune to polymorph effects) into specific groups of creatures/objects.
so, you can change an invisible stalker into a human, but it doesn't allow the reverse.