While people in Star Trek don't have "souls", they do have "neural energy" which is their brain pattern, or their consciousness. A persons physical body doesn't take up much computer storage and is apparently not much more complex than a holodeck character, as we learned in the DS9 episode Our Man Bashir:
EDDINGTON: The holosuite is specifically designed to store highly complex energy patterns. The computer's processing their physical patterns as if they were holosuite characters. Trouble is, I'm not reading any neural energy.
ROM: Neural energy has to be stored at the quantum level. The holosuite can't handle that.
ODO: So if their physical bodies are stored here, where are their brain patterns?
QUARK: Everywhere else. Their brain patterns are so large that they're taking up every bit of computer memory on the station. Replicator memory, weapons, life supports.
Neural energy is elaborated on in the TNG episode Times Arrow pt. 2:
CRUSHER: The cerebellum, the cerebral cortex, the brain stem, the entire nervous system has been depleted of electrochemical energy. Here's another one. Same neural depletion. These people did not die of cholera. They died because their neural energy was drained somehow.
Now in terms of matter transportation a physical body and its electrochemical energy should be no different, but apparently they do separate them. The neural energy seems to be the most important part to keep the persons consciousness intact
In TNG's The Lonely Among Us Picard was inhabited by an energy being and both the being and Picard transported off the ship, keeping Picard in a pure energy state. Afterwards the being left Picard and the following happened:
TROI: Commander, wait! It's the Captain. But only the Captain. He's out there alone!
RIKER: The entity, has it abandoned him?
TROI: No, but the combination wasn't possible out there. He's in trouble, sir. We have to beam him back.
RIKER: Beam him back as what? He's nothing but energy now.
...........
DATA: I knew we had to have the Captain's physical pattern here, sir. He was the last one to beam out.
RIKER: Is what you're thinking possible?
DATA: Unknown at this time, sir. I hope the Captain remembers his physical pattern is here. If he has, his energy has moved into the transporter relays by now.
RIKER: I wish we had some sign that he's in here. I guess we have no choice but to risk it.
DATA: Energising, sir.
(Slowly, Picard materialises on the platform)
PICARD: What the devil am I doing here?
RIKER: Sounds like our Captain.
TROI: But confused. This Picard pattern was formed before he went out there.
PICARD: What's happening to me, Number One? I was preparing to beam out to somewhere. And I remember there was talk of an entity? But it all seems so vague.
DATA: I believe the Captain is now his separate self, sir. Much of what happened is naturally missing.
So in this instance Picards physical pattern was still stored in the computer just hunky dory, but what I can only presume was his neural energy was kicking about on its own and was reformed with his body. The neural energy is what's key, otherwise you just get a slab of dead meat.
It's also worth pointing out that we've seen someone get transported from their own perspective when Barclay was transported in the TNG episode Realm of Fear. I couldn't find a full video of it, but we see a bit of it in this video at 5:23:
From Barclay's perspective it was a continuous process with no moment of unconsciousness.
Then there's ENT's Daedalus where the subject of transporter copies was directly addressed, this is from the father of the transporter:
EMORY: All breakthroughs are hard to imagine before they happen. When I developed the transporter, most people simply couldn't grasp it. Some still can't.
ARCHER: I have to confess, given a choice, I'd much rather use a good old-fashioned shuttlepod.
EMORY: I'll never forget the protests when the transporter was first approved for bio-matter.
DANICA: Oh, God. Here we go.
EMORY: People said it was unsafe, that it caused brain cancer, psychosis, and even sleep disorders. And then there was all that metaphysical chatter about whether or not the person who arrived after the transport was the same person who left, and not some weird copy.
TUCKER: Which would make all of us copies.
EMORY: I had to fight all of that nonsense, and I'm not going to tell you there weren't costs. I'm living proof of that, but I won. Mankind is better off. Makes everything I've fought for worthwhile.
So it seems to be accepted that the transporter does not create copies, but in some outlier circumstances duplicates can be made. Thomas Riker was only made because there was a unique storm that was able to keep the second transporter pattern stable and cause it to rematerialize, as we learn in TNG's Second Chances:
LAFORGE: Apparently there was a massive energy surge in the distortion field around the planet just at the moment you tried to beam out. The Transporter Chief tried to compensate by initiating a second containment beam.
DATA: An interesting approach. He must have been planning to reintegrate the two patterns in the transport buffer.
LAFORGE: Actually, it wasn't really necessary. Commander Riker's pattern maintained its integrity with just the one containment beam. He made it back to the ship just fine.
CRUSHER: What happened to the second beam?
LAFORGE: The Transporter Chief shut it down, but somehow it was reflected back to the surface.
PICARD: And another William Riker materialised there.
RIKER: How was the second pattern able to maintained its integrity?
LAFORGE: The containment beam must have had the exact same phase differential as the distortion field.
RIKER: Which one of them is real?
LAFORGE: That's the thing. Both. You were both materialised from a complete pattern.
CRUSHER: Up until that moment, you were the same person.
And just for the heck of it, here's the dialogue from the TNG episode Unnatural Selection the OP refers to:
PICARD: Now, quickly. If the changes in Doctor Pulaski's DNA were reversed, would it be possible
DATA: It is not reversible, sir. The subtle molecular transposition of
PICARD: Yes, yes, yes. But say if it were undone, would she be normal again?
DATA: As normal as ever, sir.
PICARD: You said that the transporter could be modified to filter out the changes in Doctor Pulaski?
O'BRIEN: Yes, sir, but we were unable to locate her trace pattern.
PICARD: Well, what if we used a sample of her DNA, say from a blood test taken before she was exposed to the disease? Could that be used to filter out the genetic changes?
O'BRIEN: Well, I'd have to get into the biofilter bus to patch in a molecular matrix reader. That's no problem. But the waveform modulator will be overloaded without the regeneration limiter in the first stage circuit.
DATA: Interesting. However, theoretically,
PICARD: Data.
DATA: Yes, sir.
PICARD: Can you do the modifications?
O'BRIEN: I think so, sir.
PICARD: Then make it so.
O'BRIEN: You heard what he said. Let's get those panels off.
[Doctor's office]
RIKER: A blood test, a tissue sample, anything that would have a sample of Doctor Pulaski's original DNA.
DATA: No, sir. Her records were shipped by way of Starfleet headquarters. They have not caught up with us yet.
RIKER: This is ridiculous. A cell, a single cell. Let's check her quarters.
[Pulaski's quarters]
RIKER: Anything. A fingernail, a hair.
DATA: Hair brush.
(Which for some reason she keeps at the back of a clothes drawer)
DATA: It has a follicle, sir. Live cells.
[Bridge]
PICARD: Darwin Station, this is the Enterprise.
PULASKI [on viewscreen]: Go ahead, Captain. I'm here.
PICARD: Doctor, we may have a solution. We have a sample of your normal DNA to use as a filter in the transporter. We think that we can beam you aboard while filtering out any of the genetic problems caused by the disease.
PULASKI [on viewscreen]: Interesting theory, Captain. If it works, we could use the same technique to save Doctor Kingsley and her colleagues.
PICARD: I think you should know, this has never been attempted before.
PULASKI [on viewscreen]: Well, I'll tell you one thing. If I live through this, I'll have a much better understanding of geriatrics.
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