I recall discussing this with you in Budapest last summer, and I was curious about the same thing. Here are my thoughts a year later:

I haven't seen any computational examples in the literature (not that they don't exist), for Q1, but in my experience any computation using Bar-Natan's formal-pictures theory (dotted cobordism version) can be translated to Khovanov's $\mathcal{H}^n$-theory. The module associated to a tangle is the projective $\mathcal{H}^n$-module whose generators are the generators of the Bar-Natan complex; in other words, you just view a Bar-Natan generator as representing all possible outside closures (and choices of signs on the resulting circles). The differential is given by applying the usual 2d TQFT to the (closed-off in all possible ways) cobordisms in Bar-Natan's complex. When you make a simplification in Bar-Natan's complex, e.g. a Gaussian elimination of two generators $x$ and $y$ connected by an identity-cobordism component of the differential, there's a corresponding homotopy equivalence of the associated $\mathcal{H}^n$-modules.

There are tons of computations using Bar-Natan's theory, so if you want to see how an $\mathcal{H}^n$ computation works, you could probably take one of those and just follow through in $\mathcal{H}^n$-language. (There are probably equally good examples, but I can't resist linking to my own computations for pretzel knots that I was working on last summer, arXiv:1303.3303- you could probably take any part of that computation and apply the $\mathcal{H^n}$ functor to get a computation in that language.)

For Q2, there's a whole lot on the representation theory side of the picture. Stroppel's paper arXiv:math/0608234 (and several other related papers) present $\mathcal{H}^n$ as an idempotent truncation of a larger, quasi-hereditary algebra $A_{n,n}$, which is (if I'm saying it correctly) the endomorphism algebra of a certain block of a parabolic category $\mathcal{O}$ for $\mathfrak{gl}_{2n}$. There's a corresponding tangle theory that works over $A_{n,n}$ instead of $\mathcal{H^n}$ and still recovers Khovanov homology. The category in question is equivalent to a category of perverse sheaves on the Grassmannian $G(n,n)$, but now I'm way out of my depth! This material might be an interesting read, though, if you're looking for how $\mathcal{H^n}$ fits into the broader scheme of things.