You have to realize that Prophet
is the suit, not a person. Laurence Barnes (aka Prophet) was a person, Alcatraz was a person. But by the time Laurence shot himself - right at the beginning of C2 - the suit had already "downloaded" his memories and psychology into the suit and continued to believe it was Prophet even after Alcatraz was put into the suit, constantly cramming the "old" Prophet's memories and identity into Alcatraz until eventually he also began to believe he was Prophet. But Prophet is really nothing more than a ghost in the shell, a symbiotic AI that believes it's a person.
Alcatraz was already badly wounded at the beginning of C2, almost near death when Prophet put him in there...the game's reason for keeping him a silent protagonist was based on his vocal chords being destroyed. At some point the suit figured out how to speak while being in stasis for so many years (as you said, terribly explained!) and I believe by that time Alcatraz's body was completely irrelevant, nothing more than a mush of bones and muscle and the suit (i.e. Prophet) had fully taken over and established an identity.
Whatever identity/memories that Alcatraz himself had were lost over the course of C2 and the stasis period, just like the suit he became an empty shell which began to believe it was Prophet.
Laurence is dead, whatever was left of Alcatraz is also "dead" in the suit, there's no reason to call him Alcatraz anymore, it's just an AI-infused brain for all purposes. That's why characters through the story keep saying "what the hell are you, are you even human?".
About the dopamine receptors...I have no idea what they were talking about. I'm going to assume the brain remaining in the suit was still left intact because that's where dopamine receptors are located, and I guess when you're put in the suit your receptors get blocked by default. Even though by that point the suit would have heavily merged with the brain.
As you said it's terribly explained, I'm fine with writers using the element of "mystery", but when they leave HUGE holes that leaves fans trying to figure out what the hell is going on it's pretty poor narrative/story-telling. Even what I've said so far is a rough guess.