This isn't a existential crisis at all. This is merely playing with what is "unknown" to people.
In the first part, it is arguing about "what is the real you?". Our answers reflect on our flawed conception of who we are. We think we are more than the tools we use to experience (bodies, DNA, etc) and our experiences, and therefore, we believe when those things when removed, won't change us. These experiments shows us that our ideas break apart, because we haven't figured it out yet for sure about where the boundaries lie. Shows our limits of knowledge, more than anything else.
The teleporter experiment is just ridiculous, because it assumes it is even possible to replicate you as you are. This is far from possible as far as we know. We have tried cloning things, and even those clones are not the same. So this is based on a failed assumption to begin, and until this is actually possible (which it won't be), then how can one even create an existential crisis from it?
The split brain experiment fails because it tries to separate some part of you, and assumes that changes you. It doesn't, because your body compensates and works with what you have. Bob is not going to be you, because his body is going to compensate and over-ride everything that is you. No it not going to be possible for Bob to read your memories, because frankly even though we are all similar, we all are distinct, and therefore our brains are not going to be compatible enough to share information like that.
Cell replacement test - this already happens with your body. You are your body + your experience. It doesn't matter whether your body is replaced by metal or by an identical cell. It is merely a tool for you to experience, and doesn't have to be exactly the same.
Body scattering - this kills you obviously. If you are not able to provide nutrition to your cells within a small amount of time, they all die, and so do you. So if you scatter and get back together before any important ones die, then you are you.
The continuity part that he uses to later argue about souls, is basically again your experiences. If you don't get to continue your experiences to the old age, then that is not you. You are basically here to know what it is to be you. That's your purpose really.
So how to get past all of this? The reason this plays with our minds (if it does for anyone of you), is because you believe you are more than your body/dna and your experiences combined. If you choose to understand that you are not, all these experiments fail on their own.