Good observations.
If the technology ever develops so that a brain could be attached to the nervous system of a different body (which is by no means even plausible, at this point) it is still possible that the brain would have to learn to "recognize" the new web of nerve endings in a long process not unlike what a newborn goes through.
It takes human infants many months to develop the proprioceptive sense, through which we perceive the movements of our body in space.
After such a process in a new body, could we say for certain that the "owner" of the brain would be the same person that we knew in the old body? Would they not have to be profoundly changed?
And don't forget, the brain ages, too, and placing it in a new body wouldn't necessarily have any effect on that process. It would be sad to see Alzheimer's overtake someone who had just had their brain placed in a 30-year-old body.

Thanks Ron. I have generally accepted the brain-seat hypothesis, but not spent much time on thinking through its ramifications, the body-transplant being problematic by opening the door to brain (and therefore personal) "immortality", or at least unnaturally long life. Even more frightful is the idea that somehow our brain (the germaine parts) might be "downloaded" into a purely artificial environment. I.e., is personhood in the brain matter OR in the information encoded therein? If the latter, which seems likely, any Christian theology based on this materialist reduction is in REAL trouble.
I'm not convinced that there necessarily IS (or need be) a way out of the conundrum. I have to wonder what relevance the quad-gametic (chimeric) human might present. In this case, you start out with two distinct individuals (early zygotes) who, prior to their being ever perceived or socialized as two individuals, merge to form one ontological being (merely with two different DNA patterns). It is possible that our body-transplant survivor becomes morally something similar to the chimera, a perfectly redeemable individual free moral agent BUT one that is in essence neither of the men who entered the hospital, but a new one. The only REAL problem is that you have Mr. A's and Mr. B's relatives (and possibly the new Mr. A+B himself) claiming they know who he is by what he was.
Interesting problem... but for heads wiser than my own.

Steve,
As I said, I'm sympathetic to your concern. But I think that it's tough to figure out how to make your argument work in real life.
Once you have admitted that "heart, blood, hand, cornea, liver, lung, skin, &c. transplants all seem perfectly licit," you now have to find a point to draw the line. At what point are you transplanting "too much"? Jimmy points out the possibility of switching heads. Or what if the injury is such that B only needed A's body from the diaphram down (that is, B would keep his own head, arms, and chest cavity, but acquire A's stomach cavity, legs, etc.)?
For the metaphysicians out there, this is basically a variation on the "Ship of Theseus" problem.
I think that, ultimately, "the brain as the seat of personality" is going to make the most sense as the way out of this conundrum.

As for baptism question, I would think that baptism applies to the person, not the body. If we accept that a body transplant is possible, and that the personality will move with the brain, then if B was baptized, the hybrid of A's body and B's brain will be baptized.
Yes, so the theory would go. BUT, we baptize whole bodies! That is to say, we baptize body and soul. Baptism is a mark on the soul but provides grace for the redemption (and eventual resurrection) of the entire body. I am doubtful that there really IS a particular seat of the soul or personality or innermost man. The human person, as an object of redemption, IS an embodied soul and not a person without a body. That being said, heart, blood, hand, cornea, liver, lung, skin, &c. transplants all seem perfectly licit. But, because a person IS a body as well as a thing that is the supposed seat of personality, a whole body transplant would NOT be licit. And even tho' I have no theory to offer to justify this, I'm nevertheless convinced it's true. My theory, if it exists, would involve essence of the person and it not being materially (or materialistically) reducible to mere brain or mere "seat of personhood" or mere body. But that theory does not yet exist, at least in my brain ;-)
Cheers!

I just know that this is going to end up with heads living in jars, like on Futurama.
"I am not a crook's head." - Ricahrd Nixon

So what happens morally if technology someday allows us to download our personality into a machine so that we can "live forever"?

I do agree with you that the identification of personality with the brain at the expense of the body is at least problematic.
However, it seems to me that if we take Biblical language about the heart literally, then we would have to say that heart transplants were problematic. I think it's a worthwhile point to raise, but almost certainly not decisive.
As for baptism question, I would think that baptism applies to the person, not the body. If we accept that a body transplant is possible, and that the personality will move with the brain, then if B was baptized, the hybrid of A's body and B's brain will be baptized.
I think it's clear that the hybrid of A's body and B's brain would go home to B's wife. But this is only one of the many "personal and social identity issues" that I think this kind of surgery raises.
I agree with Jimmy that this sort of scenario should raise serious worries about brain death as an adequate criterion for somatic death.
One further distinction to draw is that, supposing organ transplantation had been possible 100 years ago, it would seem to have been licit to harvest organs from an individual who had gone into cardiac arrest, even though at present, we know how to revive patients after cardiac arrest. The reason is that medicine at that time lacked the technology to revive cardiac arrest, and so the fact that we now have this technology would not have helped patients who went into cardiac arrest then.
However, suppose that you are involved in a car accident tomorrow which causes brain death, but does not cause bodily death. In this case, you can be kept alive, and there is the possibility that a few years down the road, medicine will discover a way to reverse the brain injury. Thus, in cases of brain death, the possiblity of future medical advances is a real issue, in a way that it was not for the cardiac arrest definition of death.
In other words, Jimmy's focus on total system failure is morally significant because only total system failure is certainly irreversible. As long as the total system remains functional, even in some seriously impaired way, the possibility exists that some future advance in medical technology will enable doctors to restore a failed subsystem.
- Ron

There is an old made for TV movie with just that premise--"Who is Julia?"

Isn't this idea, which if I'm reading right is, that personhood resides in one's brain, dangerously too Greek, and contraindicated by Scripture itself? Scripture points variously to the bowels and the heart (and possibly bones and belly) as the seat of man's innermost being. Who are we to suggest that modern medical science has finally "found" that the "true seat" is in fact the brain? More importantly, the NT speaks clearly of the resurrection of the body (in the manner of Christ's resurrection) being believers' blessed hope. Christians do not merely believe in the immortality of the soul, but in the resurrection of the body, to which the former is mere corollary.
I can offer no alternative theory. But I'm not satisfied that a full body transplant would be licit, not especially on the dubious basis that the brain is the seat of man's innermost being. Whose wife would the recovered patient go home with, especially if the transplant trauma has removed much or all of the brain's former memories? Would B's baptism apply to the previously unbaptized A's body? Or would the now "dead" A's baptism apply to the previously unbaptized B's brain?
I'm not sure where the line lies, but it seems certain that body transplants are a bit over it. Nevertheless it is good to get this stuff mulled over, for it seems almost certain that, licit or not, such a procedure will one day occur... and the Church ought to have a redemptive answer.