For smooth pasting, see: Seydel, Rüdiger, and Rudiger Seydel. Tools for computational finance. Vol. 3. Berlin: Springer, 2006. The author deals with American options therein, but the technique works more broadly on appropriate $\varphi$.

Also en.wikipedia.org/wiki/Variational_inequality. What in particular do you want to know about this obstacle problem?

I have not; I was just curious. Sometimes (though not always, or even often) having intuition into the origins of the problem helps.

Out of curiosity, how did you come to this PDE? (also, I assume $\partial_s$ should be $\partial_s f$)