My mats are the thick heavy ones (in a two horse slant) and came with the trailer - they're the same weight as my stall mats. Fortunately, they're in three sections albeit snuggly fit. In my old trailer, the mats were lighter and also fit snug. If they didn't snug, I'm sure they would have been all over the place. I think a lighter weight would work for you only if you can get a good close fit.
For my wood floor, we used Thomson's Water Sealant to protect the wood - easy to use as it just paints on. As one of the other posters mentioned, the little bit of gapping that happens between the boards does help diminish moisture build up.
When we have to move mats, either the stall or trailer ones, we use vice grips and some baler twine and that makes them alot easier to manouver. The baler twine is tied in a big loop and goes in the jaws of the vice grip and then the vice is clamped on the mat - after that you can straighten up your back and lift and tug the mats around with relative ease (you'll need gloves otherwise the twine digs into your hands when your pulling).