NEW ANSWER:

Yes there exist infinite Morse index solutions in all dimensions $N\geq 3$. For example you can take the solution in the Li Chen paper
$$
\phi(x,y) = \frac{\ln(32)}{(4+|(x,y)|^2)^2}
$$
and trivially cross with $\mathbb{R}$ to define 
$$
\phi(x,y,z) = \frac{\ln(32)}{(4+|(x,y)|^2)^2}
$$
Clearly this continues to solve the PDE on $\mathbb{R}^3$. Moreover, by the second link (Dancer--Farina) it cannot have finite Morse index. This works for all $N\geq 3$. 

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EDIT: As pointed out by Willie Wong, the original answer considers the wrong question.

There are finite Morse index solutions for $N\geq 10$. See remark 1(i) in the second paper you reference. 

>> [...] for every N ≥ 10 the equation (1.1) possesses a radial stable solution. The
existence of such a solution is a consequence of the analysis performed in [12], as
was remarked in [6].


(For your second request you should be a bit more specific exactly what you are interested in.)