The thing that's weird about this galaxy is that some of the stars go around the core clockwise, and some anti-clockwise. Apparently, this isn't problematic for the stars in the least -- as the article points out, there's about seven orders of magnitude difference between the diameter of a star and the average spacing, so collisions between them are quite rare indeed.
Also, the typical rotational velocity of stars is a few hundred km/s, while the nearest star to the sun is about 4*10^13 km away. That's somewhere on the order of a thousand years or two to cover that distance.
So, suppose that a star has to come within the average distance of Pluto's orbit to count as an "event". (The mean orbital velocity of Pluto is around 5 km/s, so this is nowhere near close enough for a capturing event, or even much of a deflection.) That's 1*10^10 km. That means that a given star will experience an event once every 10^10 or so years, to an order of magnitude -- in other words, about twice the age of the sun, or less. So fewer than half of the stars will have ever had an event of that sort. Most of them will probably have had a star come within a few pluto-distances away. I don't know how likely that is to affect solar-system formation, but given the considerable difference between passing velocity and rotational velocities of the disks, I doubt it will affect them much.
Constellations get to be interesting, I think. Half the stars in the sky don't move very much at all, because they're co-rotating. Of the other half, most of them are fairly far away, and moving slowly. There'll be a small handful of stars (three or four, average), which move about 90 degrees around the sky in a thousand years or two. That's about a degree every dozen years. A couple of hundred years to get from one end of the big dipper to the other.
That's an interesting time period. My guess is that it's not enough that an individual person is likely to notice (though maybe it would be, if one of the moving stars was especially bright, and/or very close to another star), and I'm not sure how much it would be noticed in oral records, but I suspect it would be quite notable as soon as a culture started making star maps. At some point, people start noticing that stars are mostly moving in certain parts of the sky, and in certain directions -- there's a part of the sky where some stars come from, and a part where they are going.
That's going to do really interesting things in the cultural myths, I'd imagine.