There is a nice tool in representation theory, the Howe duality, which as I know works for certain pairs of classical Lie algebras (the reference to the complete list of Howe dual pairs is appreciated very much). Is there any extension of the Howe duality for exceptional algebras?
There is a theory of dual reductive pairs and examples for exceptional Lie algebras. For example, in $E_8$ we have dual reductive pairs $(A_1,E_7)$, $(A_2,E_6)$, $(G_2,F_4)$, $(D_4,D_4)$. These are used implicitly in constructions of $E_8$; for example $(G_2,F_4)$ corresponds to the Freudenthal-Tits construction and $(D_4,D_4)$ corresponds to the triality construction.
A reference is:
MR1264015 (95b:17010) Rubenthaler, Hubert
These seem to be of interest from the point of view of automorphic forms which I know nothing about.
Yes, and it was studied both from the point of view of the local theory (correspondence of the infinitesimal characters) and automorphic forms. The following review of Jian-Shu Li's paper contains a nice selection of references:
MR1682229 (2000b:22014) Li, Jian-Shu, The correspondences of infinitesimal characters for reductive dual pairs in simple Lie groups. Duke Math. J. 97 (1999), no. 2, 347--377.
More recent work was done by Gordan Savin and Wee Teck Gan.