Simple enough:

Run it through the die or not?

Look at it this way; We want our loads to be as consistant as possible. We don't really know what dimensions the factory sized those cases to nor if there are any variations in them. We have the tools to FL size them, it's easy to do, it doesn't cost anything and it doesn't take long to do it. You would likely have them sized in less time that it took to get to your computer and ask the question, certainly less time than it's taken to wait for a response.

There is no down-side to sizing them, there may be to not doing so.

Short answer: No. Drop a couple in your chamber to make sure they'll fit (I've never had a standard case not fit a factory rifle), then bump the resizer ball of the resizing die thru the neck to make sure they're round, chamfer the mouth, prime, charge, and shoot. If you get any that diverge greatly form the group, toss the case in the brass bucket and go on.
THEN, once the brass is 1X, you can trim to a uniform length and do all the other neat things to prep your brass for max accuracy.

I'm with stillbeeman. Get the neck rounded out, chamfer, load and shoot. Since the brass is already at min spec running through the fl die will not change much. Do your better prepping after fired and mated to your chamber.
If you have a gun capable of .75moa accuracy it can do it with just this little bit of prep on virgin factory cases.

Most factory brass should be neck sized due to dents and other irrgularities in the necks. I've found that some high quality brass (Lapua, sometimes Norma) is near-perfect out of the box.

no

I DO

I only run the new brass through the FL die enough to round out any dents in the case mouth.I also trim and chamfer to get every case the same.

Yes. It should be at least neck sized because many samples of new brass have necks that are mis-sized or not round.