For example, does a PC Kitsune only have the ability to Change Shape for one minute at a time at first level? Or is it permanent until another Change Shape occurs?
To Answer this question just need to look in the Bestiary (p. 298) under the Universal Monster Rules.
A creature with this special quality has the ability to assume the appearance of a specific creature or type of creature (usually a humanoid), but retains most of its own physical qualities. The creature cannot change shape to a form more than one size category smaller or larger than its original form. This ability functions as a polymorph spell, the type of which is listed in the creature’s description, but the creature does not adjust its ability scores (although it gains any other abilities of the creature it mimics). Unless otherwise stated, it can remain in an alternate form indefinitely. Some creatures, such as lycanthropes, can transform into unique forms with special modifiers and abilities. These creatures do adjust their ability scores, as noted in their descriptions.
Format: change shape (wolf, beast shape I); Location: SQ, and in Special Abilities for creatures with a unique listing.
As you can see it specifically states here that change shape is duration is indefinite, and the entry under Kitsune does not counter this.
It lasts as long as the spell does unless the ability specifically specifies otherwise.
Some 3.X/PF abilities can be frustratingly vague when it comes to their wording, but for the most part you'll see things laid out in fairly clear black-and-white in cases like this. So, here, the ability is worded as follows:
"Change Shape (Su) A kitsune can assume the appearance of a specific single human form of the same sex. The kitsune always takes this specific form when she uses this ability. A kitsune in human form cannot use her bite attack, but gains a +10 racial bonus on Disguise checks made to appear human. Changing shape is a standard action. This ability otherwise functions as alter self, except that the kitsune does not adjust her ability scores."
Which means that indeed the Kitsune obeys the duration of the spell, since no other duration is given. However, this ability is Supernatural, rather than Spell-Like (though I can't imagine why - Paizo's mess, though, not mine), so UNLIKE Alter Self it cannot be dispelled, nor can it be counterspelled. It's still subject to antimagic fields.
Quite often when there's a supernatural ability listed, it will include a line like "as cast by a N level" or "using their character level" or something along those lines. For all intents and purposes, they're identical to the spell in range, duration, etc except where explicitly listed otherwise and they can't be dispelled or countered.
In this case, the ability description says "This ability otherwise functions as alter self, except that the kitsune does not adjust her ability scores" and that spell specifically notes duration as one minute/level.
This depends very much on the specific wording of the ability, but unless a duration is mentioned within the text of the supernatural ability, the ability is permanent. In the case of the Kitsune, the player can stay in their changed shape for as long as they want. The line about the ability working like alter self is only there in case a situation comes up where you need to know more specifics of the ability, and how other spells interact with it.
Note: Typically, at-will abilities with a duration of 1 minute or longer can be assumed to be always on, or permanent. If you have unlimited uses of an ability, you can just re-use it again before the duration runs out to extend the duration. The new, longer duration overlaps the previous, nearly expired one.