You might also want to include some symbols like the symmetric difference, relative complement and aleph-null.
Has anyone else seen I used for irrationals?
No, but neither have I seen "I" represent the "set of imaginary numbers".
There does not appear to be a regularly used symbol for the irrationals.
I believe the standard is R-Q. Q with a horizontal line on it, or R\Q.
(Problem) At the end of the first chapter and beyond, the book is slightly over my head. I only cover a few pages a day at my current pace because I have to really think about the material and I have other responsibilities. Sometimes I need to look up other references to understand certain things. At my current pace it will take me 10-12 months to finish this book. Is the payoff from reading and understanding this book worth the effort or should I read other books?
The first chapter does not even require much algebra. I think the issue is that you haven't adapted to the style of writing in higher level maths books (although this is really nothing compared to something like Rudin). You'll have to get used to re-reading multiple times and filling in the gaps on your own. Maybe, as I said before, you should supplement it with Apostol to help you.
This book is certainly worth it, no matter how long it takes. It sets you up perfectly for a rigorous multivariate calculus course and real analysis.