Would it be possible for aliens that originated and evolved in a planetary system beyond the ring singularity of a rotating black hole to send individuals out into a normal universe to colonise it? Or would the different physics disrupt their biological processes too much for them to survive or reproduce?
There is no particular reason to think that the region beyond the disk of a ring singularity must differ in its physics in any way other than the incidental shape of spacetime, and associated distribution of mass and energy, producing a naked singularity.
On the scale of biochemistry, that larger-scale structure of space is pretty much irrelevant. If the ring singularity is large enough that you can pass through it without being torn apart, and if you can somehow survive the trip across the inner horizon that exists on "our" side of the ring singularity, then yes--the new universe that your aliens end up in by virtue of heading for the inner horizon and allowing themselves to be ejected from the corresponding white hole would likely be perfectly habitable.
This is in fact the mechanism used in Stephen Baxter's Xeelee Sequence to allow baryonic life to escape this universe and continue in others--the Xeelee construct a naked ring singularity in our universe, allowing access to other universes via the Kerr white hole that forms on the other side.
Stable orbits in the ergosphere of a Kerr black hole are possible. The users in one of the sister sites have some maths showing how.
Inside a Cauchy horizon, things would even be spacetimely chill - you wouldn't be forced to necessarily move through time-like dimensions only.
Once you have planets, and an energy source (infalling radiation from outside the black hole, or starts within the ergosphere might do), then yes you can have life as we know it. Getting out of the black hole would then require magic; not mystic magic, but magic in the sense of technology so advanced that we 21st century denizens wouldn't be able to comprehend it.
However, and this is the sad part: Penrose himself had something to say in his research, and the Wikipedia article on ring singularities reads thus:
It is generally expected that since the usual collapse to a point singularity under general relativity involves arbitrarily dense conditions, quantum effects may become significant and prevent the singularity forming ("quantum fuzz"). Without quantum gravitational effects, there is good reason to suspect that the interior geometry of a rotating black hole is not the Kerr geometry. The inner event horizon of the Kerr geometry is probably not stable, due to the infinite blue-shifting of infalling radiation.
So it just might be that ring singularities might belong in the theoretical realm only.
If that doesn't stop you, then by all means do it like Marvel does and write awesome stories:
Titans act as the "masters" of the techno-organic beings known as Phalanx. They are singularities of consciousness so vast and dense, that they have caved in on their own combined intelligence to form black holes.