Not sure if it's just citing the MM, but Rules Compendium describes slam attacks (under Natural Attacks on pg 100) as "batters opponents with an appendage," which lends strength to the "must have the appendage free to use" answer.

Then on page 150 (under creatures with swim speed for the underwater combat rules), the RC mentions that some creatures have a Slam Attack that represents a Full Body Slam (like Darrin was mentioning) which is an important distinction underwater as "Appendage Slam" attacks will take attack penalties for swinging through water while a "Full Body Slam" from a creature with swim speed doesn't take the penalty.

Together, these rules seem to indicate that, unless otherwise specified in the monster entry, slam attacks are appendage attacks.

That said, I see no particular reason you couldn't Kick with your slam attack, since those are appendages that can be swung like arms and should be just as dense as the warforged's arms. I considered if maybe the ability to use kicks was a special ability of unarmed attacks that made them different from slam attacks, but several creatures have kicks and hooves as natural weapons and they don't have to stop standing on their legs to use them (even as part of a full attack).

In fact, page 100 in the rules compendium makes me think even more strongly that you could use kicks based on how its described in the paragraph on Combining Weapons. When using both Manufactured Weapons and Natural Attacks together in a Full Attack, you basically use the Manufactured Weapon as the Primary Weapon with no penalties, so one attack at highest BAB, then follow up with any or all of your natural weapons as secondary natural attacks (even with the natural weapon that is normally your primary natural weapon). Seems *to me* like it should be fair game here to Two Hand a Greatsword and use a Slam as a Kick as a secondary natural weapon (-5 to attack with half strength bonus to damage).

So, my answers based on RC:

1. Yes. Based on Thurbane's reference to the FAQ, I'd say you could either momentarily hold the Manufactured Weapon in one hand (if you're not using it to attack) and use your free hand to slam (or claw) OR you could probably kick without letting go (but it'd be good to verify with your DM to see if they agree with this ruling).
2. Yes, but you won't be able to free your hands without dropping something as a free action or sheathing it as a move, so it's kicking or disarming one hand.
3. Nothing in Eldritch Claws actually suggests you must Replace your hands with claws. I'd say, "why not both?" Two Hand a Greatsword, power up the eldritch claws, and follow up with a Slam Kick when the occasion calls for it.

Interestingly, RC adds (on page 16) that Natural Weapons cannot be used to make Unarmed Attacks, so a creature with a Slam Attack arguably can't make unarmed attacks unless they are using part of their body that specifically isn't part of their slam attack. Since Slam attacks are vaguely defined as appendage attacks, it'd be hard to consider a warforged's punch to be a slam while their kick or body check is only unarmed. If the idea is that they get a slam because they're made of metal, then basically they just don't get unarmed attacks whatsoever, because their entire body already is "armed" and causes lethal damage upon collision. That said, they still only get 1 slam attack per round no matter which part of their body they use, just like how most creatures get 1 unarmed attack despite being able to use any unoccupied part of their body. It's like they're effectively receiving the same benefit of having the Improved Unarmed Strike feat while having to utilize the Natural Weapon rules rather than Iterative attacks.