Could somebody recommend a book or an online resource where I could read up on the proof that normal CDF has no close-form expression?
2. You need to look up Liouville's Principle. you could star here and follow the links
3. Most (indefinite)integrals of elementary function have no closed form representation as a finite combination of algebraic operations and elementary functions.
It results from (or at least is closely related to) Liouville's theory. You should have a look there. I don't think it answers your precise question, but it gives precise definitions and other examples (with proof) of non elementarily integrable functions, like , which should give you an idea of the kind of arguments involved.
And there are references inside as well. For instance, if you study/work in a university, you may probably easily find the following article in your maths library:
American Mathematical Monthly Februrary 1961 "Integration" by D.G. Mead p 152-156.
You may be interested to look at differential Galois theory in you want to learn more.