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If you look at the chart at the end of the article, Mercury is every planet's nearest neighbor by their methodology.

All they're finding is "on average each planet is closer to the sun than any other planet" which isn't saying anything.

I appreciated having this pointed out to me - it's fairly clear once the point is made, but I'd never really thought about it. Of course most of the time other planets are over on the other side of the sun and very far away, but it's easy to just picture the school-style orbital map where everything is in a line together and closest approach = closest, period.

I dunno, I thought it was neat! A slight paradigm shift in how I think about the solar system.

SAME! I was waiting for this comment. To me this is result is wonderful! It's like, the physical world behaving in an intuitive, mathematical way once you realize it. I loved reading this.

I love it.

and is there no such pair of planets that revolve around the Sun while staying on the same "side" of the Sun?

The way orbits work, orbital velocity is tied to distance. So unless the planets were in the same orbit, one is going to be going faster than the other (the closer one in a radial velocity sense is moving faster, and the further one in a tangential velocity)

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