nt.number theory - What are the local Langlands conjectures nowadays, for connected reductive groups over a $p$-adic field?

First, I'd like to second the reference given by JT: David Vogan, "The local Langlands conjecture", appearing in Representation Theory of Groups and Algebras (J. Adams et al., eds. Contemporary Mathematics 145. American Mathematical Society, 1993. It can be found on Vogan's webpage

Vogan's article contains a very nice exposition of the local Langlands conjectures, and Arthur's local conjectures, and Vogan's own reformulations which I enjoy. In Conjecture 1.9, Vogan gives the local Langlands conjecture, as the OP has given it. Then, in Conjecture 1.12, Vogan gives a refinement describing L-packets, in the language of perverse sheaves (which the OP may or may not like). Adams, Barbasch, and Vogan proved this refinement for real reductive groups, and Vogan's article is certainly influenced by this.

Later, in Conjecture 4.3, Vogan gives a more detailed version of Langlands original conjectures. In Conjecture 4.15, Vogan gives a refinement, which seems equivalent to some conjectures of Arthur, though I'm not sure. This applies to most cases of interest.

To be specific, regarding $SL_2$ over a $p$-adic field, one must -- in addition to a Weil-Deligne representation $phi$ into $PGL_2(C)$ -- give an irreducible representation of the component group of the centralizer of the image of $phi$.

For example, consider an irreducible constituent of an unramified principal series of $SL_2(Q_p)$, whose Weil-Deligne representation $phi$ sends (geometric, but who cares) Frobenius to the class of a diagonal matrix $diag(-1, 1)$ in $PGL_2(C)$. Note that this matrix is centralized not only by diagonal matrices in $PGL_2(C)$, but also by the Weyl element (since we work in $PGL_2(C)$ and not just $GL_2(C)$). The centralizer of the image of $phi$ will be the group $N_{hat G}(hat T)$ normalizing a maximal torus in $hat G = PGL_2(C)$, I think. Its component group has order $2$. Since a group of order $2$ has two irreps, there are in fact two irreps of $SL_2(Q_p)$ with this Langlands parameter. This fills out the whole L-packet -- the two irreps occur as constituents in the same principal series in this case.

I think the most helpful treatment of L-packets for $SL_2$ can be found in the recent paper of Lansky and Raghuram, "Conductors and newforms for $SL(2)$", published in Pac. J. of Math, 2007. It's very explicit and considers every case thoroughly, and in a way directly relevant to modular forms. There you can find proven the statements you mention about the nontrivial L-packets being related to the two hyperspecial compact subgroups -- it's also related to the fact that "generic" has two possible meanings for $SL_2$, and representations can be generic for one orbit of character, and not for the other.