Ahhhh... welcome to the land of many answers, with none of them being wrong!
Running a skimmer during the cycle is really a personal preference. On one hand, you want organics in the tank to speed along the cycle - so skimming would seem to decrease those organics and slow the cycle. On the other hand, keeping nutrients low during cycling (but not low enough to stall the cycle) by using the skimmer makes it harder for nuisance algae to establish itself before you can get critters in there to control it.
When I cycled, I had my skimmer on 24/7. It was a good time to break it in, getting rid of the microbubbles and all, when there was nothing in the tank. I'd probably do it that way again if I was setting up a tank.
If your skimmer is skimming really wet right now (clearish liquid), yeah... it could mess with your salinity, but not that much. Just keep an eye on it, and adjust accordingly. Or just tune your skimmer so it skims a little dryer.