Physical properties like DNA are one of them, and physical changes and stored memories are another. I don't feel the need to invent some abstract concept of a soul to think of myself as unique. You use words like "conscious being" that just don't have that much different connotation than religious ideas of a soul. I am a being that happens to presently be in a state of consciousness. Eventually I'll become a being that happens to be in a state of death. For me it need not be anymore complicated.
Descartes isn't defining a "soul", he is defining the basis of knowledge which requires an assumption of a state of existence.
Russell's Teapot says I feel no obligation to even try.
Regardless of whatever chances you think of some random person existing are, the chance of you in particular existing is pretty much 100% unless I'm incredibly high or in some other form of psychosis and just imagined you. Assuming this not to be the case and building on the fact that you're replying, I'd go one step farther and say your chance of being conscious is also 100%. To work off your lottery analogy, if you hold the ticket with the winning numbers, your chance of winning is 100%, not 5.7 e-11%. Now if you want to play the game of the monkey banging on a typewriter writing War and Peace, "you" might be physically identical to you, but you would be "reincarnated" in the same sense as a recycled ball bearing.
Russell's Teapot is intended to be far fetched. That's pretty much central to the whole point. Asserting that I'm claiming any "default" is strawman. I've already told you I never said you were wrong, just that I have no reason to believe you are right. You keep confusing those two things.