The single best introduction to SDE from numerical angle is this Higham's paper. It will probably give you an approximation to answers to your three questions.
a) In finance the assets are not priced on the basis of knowing exactly their cash flows. Moreover, they're not priced on the basis of knowing their expected cash flows either. Ideally, you need to know the whole distribution of future prices. It's often expressed as a pair "risk-return". The risk part is a quantified uncertainty about the return. That's why stochastic calculus seems to fit finance applications so well, it's precisely because it appears to capture our understanding of uncertainty about the future cash flows from the assets.
The sampling would represent possible cash flow paths. Each path is a possible realization of the future. In Monte Carlo methods you explicitly sample paths, and obtain the distribution of cash flows, which allows you to price the assets.
However, under certain conditions, you can formulate and solve the SDE as partial differential equations (PDE) - non-stochastic. That's what Merton did with Black-Scholes (BS) PDE approach: he linked them to SDE. Original BS paper formulated option pricing problem as a heat transfer equation from physics.
In BS equation for an option price, you can see that there are 5 inputs: asset price, volatility, strike price, risk free return and time to maturity. Even before BS equation, these were all known to be determinants of the option prices. That's why when the paper came out it immediately made a sense to practitioners. Note, now that there's nothing about the future price of the asset. The only information about the future price is volatility, which represents the uncertainty about the future.
So, intuitively, what BS equation does is it expresses the option price as a function of the distribution of future prices, namely its standard deviation. That's how SDEs are used: you express your price through the distributions of future outcomes, and if you're lucky you solution will have something simple like the standard deviation in it.
b) Monte Carlo is used a lot, but as I wrote above if you can convert the problem into PDE, then all kinds of methods such as finite elements can be used.
c) I'm not sure there's such a book, i.e. computational with measure theory. If you're mathematician I can recommend the one I used: Shreve's text "Stochastic Calculus for Finance II: Continuous-Time Models". There's no software coming with it though, it's quite theoretical, may not work for you, if you're not strong in math.
UPDATE
I want to add a physics example to a). Look at diffusion process. You can think of a single atom's path as a single path in SDE, maybe in its Monte Carlo sampling. It's totally unpredictable. However, when you look at the diffusion of large quantities of atoms, the diffusion process is very predictable in terms of the speed with wich one material goes into another in a macro level.
