last part of technical explanation of argument 1
I cannot accept this argument. I think that you're privileging a certain aspect of water above all others in an attempt to create a "simple object" of the sort that you'd need to make this rigid designation possible. Yes, water is two hydrogens bonded covalently to an oxygen atom. From the perspective of the chemist or physicist, this is what water IS. However, there are numerous other ways in which we may engage water as we encounter it. If we disaggregate these different experiences of water, then we cannot rigidly designate water in the way that you'd like to.
ebola
No doubt we can "engage" water in several different ways, but I'm not entirely clear how this applies to the way the referent of 'water' is determined. I don't think that you want to deny that water refers at all.
If you think modal statements make sense (statements such as 'necessarily x' or 'possibly y') then the only way you can deny that water is a rigid designator (I think) is to say that it is a logical possibility for there to be water which is not H2O. If this is your objection, then I would like you to give me a counter-factual in which this is the case. If you want to deny that modal statements about the world make sense then you will have to come up with some extremely impressive arguments (and join forces with your loathed and widely discredited logical positivists!).
I don't think that Kripke (this is his theory) unnaturally tries to make water into a simple substance
ad hoc to his theory of naming so much as he looked at how we naturally seem to use expressions of water and built up his theory around it.
Go out and ask your non-philosopher friends, "Can you conceive of a situation in which you have water and not H2O?" I think they will say "no." If there is no possible situation in which water (as we use the term in the actual world) does not refer to H2O then, trivially (by definition), 'water' rigidly designates H2O.
Now, you might reply with this objection:
Objection 1: But it does seem imaginable to me that water might have turned out to be something other than H2O." Back when we first investigated the molecular structure of this clear, flavorless, life-giving substance, it could have turned out to have had some other molecular structure. Therfore, the fact that water = H2O is contingent. Hence, there are contingent identities.
Reply: The case you have described/conceived is not a case in which water was not H2O. It is a case where,
for all you knew water could have been something other than H2O. You're imagining being in the same epistemic situation that the people who discovered that water is H2O were in when they first started investigating it. All you know is that there is this clear, flavorless, etc. substance that's molecular structure is still unknown to you. Now, no doubt, it is possible that you could be in this epistemic situation, investigate the clear, flavorless substance, and find out that it's not H2O but has some other molecular structure.
But this is not a case where the stuff that we refer to as 'water' is not 'H2O'. It's a case where someone is in the same epistemic situation as we once were regarding water, investigates some other, very similar substance, and finds out that it is not H2O. It's not a case in which our water, aka H2O, is not H2O.
Notice that this "explaining away" of the illusion of contingency appeals to the difference between our sensual experience of a thing and the thing itself. It also works for heat/molecular motion. It might seem like we can imagine a possible world in which heat turned out not to be molecular motion. But what we are actually imagining is a case where we are in the same epistemic situation that we once were regarding heat, namely a situation in which something feels warm. And it then turns out that something other than molecular motion is causing this warmth. But this is not a case in which that which we call 'heat' in the actual world, turned out not to be molecular motion. It's a case where something else other than heat/molecular motion causes warmth. This is because when we use the word 'heat' we just stipulate as a matter of convention (and I really think this is what most people do) that we are talking the thing that is in fact causing this sensation of warmth in us (which turned out to be molecular motion).
So, it's necessary (true across all possible worlds) that water is H2O and heat is molecular motion because there is no possible world (try to imagine one) in which we have water and not H2O, have H2O but not water, have heat but not molecular motion, or have molecular motion but not heat.
The relevance of all this to the mental state/brain state argument is this: Any identity statement between two rigid designators is necessary. 'Pain' is a rigid deignator. 'Brain state of type p' (the brain state that allegedly is pain) is also a rigid designator. If "Pain = Brain state type p" is true, by virtue of the fact that the words both rigidly designate, then they must be necessarily true. You should not be able to imagine a case where "Pain is not brain state type p" because there is no possible case in which the one, actual thing, we us the words to refer to is not itself. But, I can imagine pains without brain states and brain states without pains (like when I imagine ghosts & zombies). Therefore "pain = brain state p" is not necessary. Therefore it is false.
You will probably want to argue that 'pain' is not a rigid designator. But can there really be a more obvious case of picking out a referent by something essential to it than with a pain? Can you have a pain that does not hurt? This seems to be pretty ridiculous. If you're not hurting (or feeling pain) then how can you be
in pain? If you want to argue that "pain is not a rigid designator" (does not pick out the same object across all possible worlds) then you will have to give me an example in which we have a pain that is not painful.
You're tempted to give the possible situation of, "When your brain is in the state that is identical with pain but you don't feel pain" aren't you?

This won't work because by doing this you concede that you can imagine the brain state without the actual thing 'pain' that we rigidly refer to in this world. If you can imagine the supposed identical brain state without pain then it's possible to have the brain state without the pain hence the two are not necessarily identical, hence not identical at all.
The other option is to deny that 'brain state p' is a rigid designator. But how could it not be? How is it possible to have a 'brain state p' that is not that actual brain state, 'brain state p'? Ask yourself the quesiton, "Could 'brain state p' not have been the brain state that it refers to in the actual world (not just have been called something different)" and the answer, I think, is no. If you want to talk about brain states in general ask, "Could a brain state not have been a brain state" and the answer again, I think, is no. If you want to deny the rigid designation you have to give a logically possible case in which a brain state of a specific type is not that type of brain state or a brain state in general is not a brain state.
And you cannot explain away the apparent contingency (and therefore falsity) of the identity statement "Pain is brain state p" the same way we explained away the illusory contingency of "water is H2O" or "heat is molecular motion." This is because in those cases we exploited the difference between the sensations we have of water and heat (colorlessness etc. in the former, and warmth in the latter) and the actual things water and heat. But in the case of pain and brain state p this is impossible. Why? Because to be in the same epistemic situation that you are when you have a pain is to have that pain! If you are feeling something that hurts, then you are in pain. So you can't say, "When you imagine pain without brain state p, what you are actually imagining is feeling something else that hurts, but is not pain/brain state p". If it doesn't hurt, it's not pain.
Anyway, thats basically the technical explanation of argument 1 in full. From your earlier post I got a hint that maybe you wanted to deny rigid designation in general. If this is the case, I will try to characterize your view of naming below and you can tell me if I have characterized it right. Once I do that, I will try to give you lots of reasons to think that Kripke's more widely accepted view of how words like 'water', 'Richard Nixon', and 'pain' work more accurately represents how people actually use the language.