I've lead far more than I've seconded. I am just seeing how the art of efficient seconding can make an experience better. I want to know do people have a system? What's the simplest, quickest, most neutral (works in most situations) way to second?

One trick is to have a extra biner on your gear sling or harness to collect all of the nuts that you clean. This speeds up the belay transitions.

Just making sure you are on the same page in terms of how to re-rack once you are anchored in.

No "right way" to do it.

For me, while the old belayer is getting ready to lead (breaking down belay, putting shoes on, rope management etc.), the second can be racking. I like draws on harness, gear on sling, so second can just clip draws on my harness while I'm doing other stuff and then hand me the gear sling re-racked.

Organization! Clean the gear and re-rack it in the same order/fashion that the leader will use it on the next pitch. If I'm not leading the next pitch, I often put gear on one sling and draws/slings on another and can just give each sling to the leader or clip them to the anchor. If I'm leading the next pitch, I try to place the cleaned gear where I will want it on the next pitch.

I like the nuts on a separate biner idea although I've never tried that.

As a second, I'll often start tearing down the lower anchor as soon as I take the leader off belay. I'll leave one solid piece in and be ready to take that out and start climbing as soon as I hear that I'm on belay.

+1^^^^^ think how you like things to come back to you when you are leading, and then try to accomplish that while you second. breaking down the anchor early so you can pull that last piece and blast once on belay really speeds things up. using the slings so you can just pass them off to the lead and then start flaking out the rope so once they are reracked you put them on belay and watch them blast out without any dead time, both leading and seconding have an art to them without any real right or wrong so figuring out whats working the smoothest between you and your partner is ultimately whats going to be the right way for you.

Shut up and hold on to your nipples!

Nah, I think there's way more to be said about competent belay technique and attentiveness for seconds than any fiddly details about cleaning. This is pretty much a non-issue in my book. Don't drop the gear? Climb quickly? What else is there to say, really?