Yes it is called Tamarak Sno-Park
Yes it is free.
Yes you can park trailers there. Unfortunately the last time I was up there the parking lot was pretty full with paving construction equipment. If you get there and this is the case then you can just drive through the gates at the end of the parking lot and find an open spot off of Rock Creek Rd.
As for safety. Its pretty safe however I'm always nervous leaving my tow rig anywhere like this. I have never been broke into but I have heard stories.