- Does anyone here agree with me that the new self resulting from a perfect copy would be different than the original self?
Clearly not. Part of the problem is that they simply disagree (if you want to get into a big thing about this, I'm sure there are several threads already out there - look for stuff about the Ship of Theseus or the "Teleporter Problem") and part of the problem is that you aren't defining your terms clearly and consistently.
"New self" isn't something you've even convinced anyone here is a valid concept, partly because you can't seem to clearly define what you mean by "self" that isn't already explained differently and better by science. Note that even if you can clearly explain it you would still have to convince people, but at least you'd have made a good start.
- Do you guys think that you experience the same thing/process that religious people call the "soul"?
- Do you guys think that you experience the same thing/process that reincarnationists think comes back to life?
As far as I'm concerned those are both the exact same question.
I don't believe that anyone has a soul. I believe that I experience roughly the same sense of consciousness and self-awareness as typical humans, whether or not they believe in souls, but I don't know that I agree that there is a specific thing that any group experiences that they claim to be the soul. They believe they have one, and they experience life, but I haven't really heard people say "There! That experience! That's what my soul feels like!"
As others have pointed out, there's a bit of a god of the gaps thing going on here. What part of our experiences isn't already accounted for? We still have a lot of things to learn about the brain, but everything we have learned about memory and personality and consciousness tells us it all comes from the meat in our skulls. What's left to be the "soul"?
If I get a traumatic brain injury, it can change my personality, erase my memories, drastically alter my decision making - so has it damaged my soul? If not, what meaningful part of "me" could possibly be contained in the soul? If the soul isn't my personality, memories, decision making, or emotions - all of which we can alter by poking at the right spots in someone's brain - then what the hell is the point of it anyway?