If your trailer has a "smart charger" in it like the
Progressive line with the "Charge Wizard" then I would leave it plugged in. Our trailer used to sit plugged in 24/7 for about 12 years with no damage and the batteries lasted around 7 or 8. All I had to do was check the water level a couple of times/year - usually without having to add any.
There are other brands that have just about the same technology also.
You could use a stand alone smart battery charger like a "Battery Tender" brand that would work fine also if your converter is not a newer model.
No need to put wood under the tires. Just inflate to normal (usually max sidewall pressure) and they should be good for the winter. If the trailer was going to sit on the dirt then I would put something under the tires.
Barney
2004 Sunnybrook Titan 30FKS TT
Hensley "Arrow" 1400# hitch (Sold)
Not towing now.
Former tow vehicles were 2016 Ram 2500 CTD, 2002 Ford F250, 7.3 PSD, 1997 Ram 2500 5.9 gas engine