Re: If Sol is orbiting the galactic core ....
Yes stuff in our galactic orbit (distance from the centre) tends to have similar orbital velocity but closer or further from the centre it doesn't; hence the spiral shape of the galactic arms.
However that is not the story here, it's about the fact that the solar system wobbles up and down through the galactic plane as it orbits, some times above the plane and sometimes below and the 35my mentioned is about the periodicity of that. Going through the plane, where matter is more concentrated, it is more probable that we will encounter more random debris (be it dark or normal matter). That appears to be the hypothesis in the article linking this wobble to the extinction events.
Another factor is that everything is on the move including the Milkyway and it happens to be moving 'northward' with the galactic plane being at more or less 90 deg. to the direction of movement, face on rather than edge on. When the solar system is above the plane (north of it) we get more cosmic ray bombardment than when below as the matter in the plane acts like a shield (current thinking). It has been hypothesised that this has caused bursts of evolutionary change as well as effecting global weather conditions. Far from proven but worthy of thought if such things interest you.
P.S. - I did tell it to my dinosaur and it said "Polly wants a cracker".