I have a pole barn with a concrete floor. I do have staining from the preservative chemicals leaching out of the tires. It might be a good idea to always put down a barrier for long term storage. With the new concrete, using a board would spread the load.
I leave the battery in the trailer and disconnect the power cord so that it is not exposed to any power issues. The battery is turned off at the disconnect switch and I connect a Battery Tender about 3 to 4 days per month. A
cord is wired to the battery to provide a quick and easy connection.