Actions of locally compact groups on the hyperfinite $II_1$ factor
The answer to (1) (for second countable groups) is yes: every locally compact second countable group $G$ admits a countinuous, faithful outer action on the hyperfinite $II_1$ factor. This is attributed to Blattner, and is stated explicitly in Proposition 1 of the following article:
R. J. Plymen. "Automorphic group representations: The hyperfinite $II_1$ factor and the Weyl algebra." Algèbres d’Opérateurs. Springer, Berlin, Heidelberg, 1979. 291-306.
The action is obtained as the Bogoliubov action associated to the direct sum of countably many copies of the left regular representation of $G$ on $L^2(G)$.
Question (3) has been addressed by Kerr, Li, and Pichot in
David Kerr, Hanfeng Li, and Mikaël Pichot. "Turbulence, representations, and trace-preserving actions." Proceedings of the London Mathematical Society 100.2 (2009): 459-484.
They show that there is no hope of classifying all such actions, up to isomorphism, in general. They make this precise by using the language of Borel equivalence relations and Hjorth's theory of turbulence.
I refer you to their paper for the precise statements and a detailed discussion of what all this entails, but here is one of their results: they show that for every countably infinite amenable group $G$, the equivalence relation of isomorphism of free actions of $G$ on $R$ is not classifiable by countable structures, i.e., there is no Baire measurable way of using (isomorphism classes of) countable structures (e.g., countable groups, countable rings) as complete invariants for isomorphism classes of these actions. In fact, what they show is even stronger than this: in the Polish space of all free actions of $G$ on $R$, every isomorphism class is meager, and every isomorphism-invariant Baire measurable assignment of (isomorphism classes of) countable structures to these actions is actually constant on a comeager set of actions.
${\bf \text{Update:}}$ My original answer to (3) above addresses the possibility of classifying actions of $G$ on $R$ up to isomorphism, instead of up to outer conjugacy. As Marcel Bischoff indicated in the comments, Oceanu studied the question of classifying actions of a discrete amenable group $G$ on $R$ up to outer conjugacy in
Ocneanu, Adrian. Actions of discrete amenable groups on von Neumann algebras. Vol. 1138. Springer, 2006.
Ocneanu proves is that any two free actions of a discrete amenable group $G$ on $R$ are outer conjugate. As Ocneanu points out, the assumption that $G$ is amenable is necessary, as shown by Jones in
Jones, Vaughan F.R. "A converse to Ocneanu's theorem." Journal of Operator Theory (1983): 61-63.
A positive answer to my question (2) is provided in the last section of Popa Takesaki "Topological structure of unitary and automorphism groups".
They show that for any locally compact group $G$, the action of $G$ on the $II_1$ factor $\{\mathit{CAR}(L^2G)\}''$ is appropriately continuous. Here, the double commutant is taken on $L^2(\mathit{CAR}(L^2G),tr)$ with respect to the unique trace on $\mathit{CAR}(L^2G)$, and $\mathit{CAR}$ is the algebra of canonical anticommutation relations (a.k.a. Clifford algebra).
Answer courtesy of Daniel Bruegmann.