A few points:

1) Very nice exposition.

2) Near eq. (4) it is claimed that one cannot compute the delta \frac{\del C}{\del S} without stochastic calculus, since S is stochastic. That doesn't strike me as correct: C is just a deterministic continuous function of S, C, K, T, t, r, sigma; and computing partial derivatives does not require stochastic calculus.

3) It captures the notion that when you hedge, you use risk-neutral probabilities.

4) Generally, in practice, BS is written as follows:

   C = df ( F N(d1) - K N(d2) ), where d1 = (ln(F/K) + 1/2 s^2)/s, d2 = d1 - s, s = sqrt(sigma^2 (T-t)), df is the discount factor, and F is the forward price of S. 

This abstracts away the whole discounting business.

Note that sigma never occurs except in the expression sigma^2 (T-t), which is dimension less, thus sigma has physical dimension 1/sqrt(year), usually ("annualised vol"). C has the same dimension as F and K.