No one knows
The Beast Strike feat is poorly written. A good idea, but poorly written. Nothing says he actually uses his claws, and as you say, a Monk can make Unarmed Strike attacks without using his hands, which usually leaves hands free for Claw attacks. My guess would be that whoever wrote the feat, however, was completely unaware of this fact.
RAW, you can use the Claw attack(s) as secondary weapons in a full-attack after all normal attacks (provided you haven’t used your hands, which a Monk need not do), and a Flurry of Blows is explicitly a form of full-attack. This seems pretty likely unintended, but nothing can really be said with absolute certainty.
I probably wouldn’t allow it. I’d sooner fix or replace the Monk class than try to “throw him a bone” by allowing weird results from a poorly-written feat.
Do they occupy the "Hands" slot, preventing simultaneous use of non-weapon magical gauntlets or gloves?
No, the gauntlets are listed as weapons and no indication is ever given that they interact with the Hands slot.
I would, in most cases, treat them similarly to shields, however: single physical item, separate mechanical stats for use as a weapon and use as non-weapon equipment. This is just the same as shields, which point out that their Enhancement bonus to AC does not apply to shield bash attacks, but shields can be enhanced separately as magical weapons in their own right, in addition to their armoring properties. The rules do not specify this, however, and by RAW the two are completely independent such that you can equip both at once.
If one is wearing (Spiked) Gauntlets and holding another non-Gauntlet weapon simultaneously, must one use a move action to "draw" and switch between the Gauntlet and held non-Gauntlet weapon?
Gauntlets are explicitly listed as “Unarmed” weapons, which implies that if you drop your weapon (a Free action) and are now unarmed, but are wearing gauntlets, you should use gauntlet damage. Absolutely no need to draw anything.
Weirdly, spiked gauntlets are listed as “Light Melee” weapons, which implies that they work more like regular weapons, including the need to draw them and the inability to use another weapon while you have a spiked gauntlet equipped and “drawn,” whatever that means. Spiked gauntlets also use a separate entry that does not reference the gauntlet in any way, so we cannot equivocate the two that way. All that said, this is ridiculous; I would just treat them just as gauntlets that do piercing damage and have slightly higher base damage. After all, that’s all they actually are.
Is each Gauntlet in a pair considered a separate weapon, such that a character wearing two of them could enchant them differently? Could you wear a Gauntlet on one hand and a Spiked Gauntlet on the other?
Yes and yes. You can also use the Two-Weapon Fighting combat maneuver with them.
Are worn (Spiked) Gauntlets considered to be "wielded" or "in hand" even when you are holding additional weapons, such that enhancements like "eager" and "warning" on the gauntlet would be in effect even while holding a different weapon?
Gauntlets, yes, spiked gauntlets per the above, not-by-RAW-but-that’s-ridiculous.
I could see houseruling to prevent things like eager and warning though. Something like “any magical weapon properties of your (spiked) gauntlet are suppressed while you are using that hand to hold another magical weapon.” It’s not in the rules though.
Does the damage from standard gauntlets improve as your unarmed strike damage increases, as with the monk class feature "Unarmed Damage"?
Very ambiguous. The text of the gauntlet is as follows:
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 question is exactly what “otherwise” refers to here. If it is just the lethal aspect of the damage, which I personally find the most grammatically sensible and the most balanced at the table, then yes, the gauntlet is literally in all ways identical to an unarmed strike, except that it deals lethal damage.
But even that leaves some concerns. For example, you cannot magically enhance your unarmed strike: if a gauntlet is in all ways identical to an unarmed strike except that it does lethal damage, does that mean you also cannot magically enhance gauntlets? Are unarmed strikes and gauntlets considered the same weapon for, e.g., Weapon Focus? Do gauntlet attacks provoke Attacks of Opportunity if you lack Improved Unarmed Strike? The term “otherwise” here is dangerously broad in scope, after all.
Another reading is that the “otherwise” refers to the entirety of the gauntlet’s statblock: the name (i.e. it is a separate weapon), the damage, and so on. This would mean that no, bonuses to/improvements of unarmed strike damage do not apply to gauntlets, Weapon Focus (unarmed strike) and Weapon Focus (gauntlet) are quite separate, and so on. But then the question becomes, “in what ways is the gauntlet like an unarmed strike? What’s left?” I can’t really think of anything.
The way I rule it is this:
I houserule it so that there is a way to magically enhance unarmed strikes, either by having the warrior be “worked on” by the crafter or by them equipping some kind of item that does it (“hand wrappings,” “gloves,” whatever).
The gauntlet and spiked gauntlet both count as an unarmed strike and work like the unarmed strike in basically every way, but the gauntlets are automatically lethal, do not provoke Attacks of Opportunity, and the spiked gauntlet’s base damage is 1d4 (this does not stack with other improvements though, so if you have 1d6 base unarmed damage, all three use 1d6). Improved Unarmed Strike also ups unarmed strike and gauntlet damage to 1d4, so the damage difference rarely comes up.
Unarmed strikes can be “dual-wielded” for Two-Weapon Fighting and you can have at least two versions of it for magically enhancing, as you would with two weapons.
I’m now going to add in that bit about eager et al. because that’s a good point. ;)
Are monks proficient in "unarmed attacks", including attacks with Gauntlets?
No, by a strict reading of the rules, monks are not proficient in Unarmed Strike, but literally everyone houserules that. Whether or not that houseruled proficiency extends to Gauntlets depends a lot on how you answer the question of “Otherwise” above, and exactly how you want your houseruled proficiency to work. Since the actual rules don‘t actually offer that proficiency we cannot judge strict RAW here.
Best Answer
As the RAW you quoted yourself indicates, the gauntlet should be considered exactly as an unarmed attack except that it deals lethal damage. The 1d3 damage is just the normal unarmed attack damage, and it is provided as a base value, thus Superior Unarmed Damage would apply so long as you meet the prerequisites. About the +5 gauntlet, I wouldn't bother... it's much easier and better to get a +5 Necklace of Natural Weapons.