[RPG] Action econothe with the Pathfinder Double-Chained Kama while its chain is extended

1. Yes, the creature makes attacks as if proficient with the single kamas used in each hand, but only because they're not actual kamas but as if they were kamas. The double-chained kama remains a double-chained kama no matter how it's used.
2. No. Proficiency with the double-chained kama doesn't grant proficiency with the kama.

The double-chained kama says that

The wielder can attack as if armed with a single kama in each hand or extend the chain to make a single reach attack. By swinging the rope, the wielder can whip the kama about to disarm or trip opponents. Furthermore, if one of the weapons is dropped, the wielder can retrieve as a free action by pulling on the chain.

When a creature possesses the feat Exotic Weapon Proficiency (double-chained kama), this gives him no other proficiency with others weapons. Luckily, the emphasis-mine as if saves the character from even needing Exotic Weapon Proficiency (kama), allowing the user to attack with a weapon that's like a kama but, which, in fact, is a double-chained kama.

First, regular gauntlets. When described as weapons, they are given this description:

Gauntlet

This metal glove lets you deal lethal damage rather than nonlethal damage with unarmed strikes. A strike with a gauntlet is otherwise considered an unarmed attack. The cost and weight given are for a single gauntlet. Medium and heavy armors (except breastplate) come with gauntlets.

Note that medium and heavy armors come with gauntlets by default, which the various armor descriptions indicate:

Banded Mail

The suit includes gauntlets.

Chainmail

The suit includes gauntlets.

Full Plate

The suit includes gauntlets, [...]

Half-Plate

The suit includes gauntlets.

Scale Mail

The suit includes gauntlets.

Splint Mail

The suit includes gauntlets.

Each of these armors indicates an arcane spell failure chance, from 25% to 35%.¹ Arcane spell failure’s definition is

Arcane Spell Failure

Armor interferes with the gestures that a spellcaster must make to cast an arcane spell that has a somatic component. Arcane spellcasters face the possibility of arcane spell failure if they’re wearing armor. Bards can wear light armor without incurring any arcane spell failure chance for their bard spells.

Casting an Arcane Spell in Armor

A character who casts an arcane spell while wearing armor must usually make an arcane spell failure roll. The number in the Arcane Spell Failure Chance column on Table: Armor and Shields is the chance that the spell fails and is ruined. If the spell lacks a somatic component, however, it can be cast with no chance of arcane spell failure.

Shields

If a character is wearing armor and using a shield, add the two numbers together to get a single arcane spell failure chance.

It seems to me that if a character using one of these armors that are assumed to include gauntlets still has a 65%-75% chance of successfully casting an arcane spell, and that the chance of failure here applies only to arcane spells, gauntlets cannot make somatic gestures impossible. Harder, perhaps; they probably figure into the relatively-high arcane spell failure chances of these armors. But not impossible.

Basically, the game simply has too many rules for determining your chance of successfully casting a spell while wearing armors that include gauntlets for gauntlets to preclude spellcasting. If they did that, all those rules wouldn’t be necessary and I would expect some kind of “N/A” entry in their ASF column, and a note in the rules for arcane spell failure.

Finally, we have the locked gauntlet:

Gauntlet, Locked

This armored gauntlet has small chains and braces that allow the wearer to attach a weapon to the gauntlet so that it cannot be dropped easily. It provides a +10 bonus on any roll made to keep from being disarmed in combat. Removing a weapon from a locked gauntlet or attaching a weapon to a locked gauntlet is a full-round action that provokes attacks of opportunity.

The price given is for a single locked gauntlet. The weight given applies only if you’re wearing a breastplate, light armor, or no armor. Otherwise, the locked gauntlet replaces a gauntlet you already have as part of the armor.

While the gauntlet is locked, you can’t use the hand wearing it for casting spells or employing skills. (You can still cast spells with somatic components, provided that your other hand is free.)

Like a normal gauntlet, a locked gauntlet lets you deal lethal damage rather than nonlethal damage with an unarmed strike.

Note how locked gauntlets specifically call out that they do not count as a free hand, but that you could still have your other hand free. This is exactly what I would have expected regular gauntlets to say if they prevented your hand from counting as free. They do not, however, say that.

Finally, actually getting to spiked gauntlets properly,

Gauntlet, Spiked

Your opponent cannot use a disarm action to disarm you of spiked gauntlets. The cost and weight given are for a single gauntlet. An attack with a spiked gauntlet is considered an armed attack.

Nothing here says that spiked gauntlets interfere with your hand any more than regular gauntlets... but then, nothing says that spiked gauntlets work “like” gauntlets or any such thing, and weapons usually do take up a hand. To that, I have previously described that situation as ridiculous, and I stand by that. Spiked gauntlets should work just like gauntlets, but with spikes (piercing damage, larger damage die), even if the rules fail to actually indicate that.

So unless your spiked gauntlet is also locked—which is probably a thing you can do—it should be free to cast spells.

1. _{Hide armor is a non-breastplate medium armor, so it may or may not have gauntlets—it doesn’t seem to have an individual entry. I’m not sure the word “gauntlet” can really apply to what are really just heavy leather gloves. Anyway, if it does that would drop the lower end of the ASF range to 20%.}