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 $N\geq 3$.
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.)