The result you cite from ErdÅ‘s-KalmÃ¡r is wrong -- I imagine you miscopied it. the mistake is in the second inequality, though, on which your result does not depend. However, your proof of "the" lower bound (you should have said "a" lower bound) is wrong, because the cited result is not strong enough to prove it. You can derive it from the Prime Number Theorem, though -- it's not wrong, just its proof. But this is a very weak bound, and I don't know why you'd use it when you already have a better one.
Your heuristic in section 2 is reasonable. I don't know how much of an error term to expect, but I expect that it's small enough that the bound in 2.1 holds for large enough numbers. Remember, though, that when intervals are short (polylogarithmic) the heuristic no longer holds -- see Maier 1985 and related literature.
Your work in section 3 is not useful without error terms. Most of the subtractions have results which are subsumed by the error term when you use the PNT. Conjecture II is still reasonable, though, since you require that the difference between a and b is >> sqrt(b) which should suffice. (This is very much conjectural, of course -- we can't even get this with RH.)
The "calculation" sections are very unusual; I assume these are approximations, despite the lack of $\approx$?
Overall the paper looks fine -- non-cranky -- but at a very elementary level.