Existence must exist at all times.

[bennyboy]

2) That is indeed possible. But is it actual? 

Also, "idea" doesn't help you much in contradicting the concept of a soul because "idea" is synonymous with the word "form", and the human "form" is synonymous with the word "soul". If the continuous and consistent "idea" of-the-human-self is passed from one moment to the next, you have just described a certain understanding of the soul. I know that isn't what you meant, but it illustrates the difficulty in accounting for identity on the one hand, and change on the other. This is the classical meaning of the principle of identity. Just something to think about in the language you may use in the future.

Well, an idea is a template which may be made manifest, right? Let's look at the sense of self. You talked about continuity-- but what about discontinuity? It seems to me that due to brain damage, some people have almost nothing in common with their former "self." Some kinds of damage even cause changes in personality and preferences. The only thing that seems to be immalleable is that one either does, or does not, have the capacity to be aware that he exists. And since one does not have this capacity during sleep, it seems to me that the identity is a collection of linguistic or symbolic impressions: "I'm this kind of guy. I believe that kind of idea." and so on. In other words, I am the part which I play, and that part is a collection of ideas about what it is supposed to be like to be a particular person.

3) If the self may be an idea, it does not mean that it is indeed an idea. 

Even so:

SUPPOSING that the "self" IS merely an idea which is passed from one moment from the next: What is doing the passing in the current moment, and what other thing does the receiving in the next moment?

Some here will certainly start gesticulating vehemently toward the brain, but I think that's not really an answer. I consider all parts of consciousness mysterious-- not so much their content, which is clearly of the brain and environment, but in their-- being rather than not being.