Are there closed-form expressions providing the VC-dimension for the multi-class case for different classifiers?
VC-dimension has multiple multiclass extensions:
pseudo dimension, Natarajan dimension, graph dimension.
See here for example:
http://math.huji.ac.il/~amitd/multiclass.pdf
http://jmlr.org/papers/volume8/guermeur07a/guermeur07a.pdf

There are also extensions to continuous-valued functions (fat-shattering dimension):
http://users.cecs.anu.edu.au/~williams/papers/P53.pdf