1. I wouldn't worry about the grass or putting anything under the trailer either. The shade from the trailer will choke out anything growing under it. If you are in a snow state, the grass will soon be dormant anyway.
2. Considering you are parking on grass and dirt, definitely put some lumber under the tires. Get something long enough and wide enough, a single board that will comfortably support your tires, like, at least a 6 foot 8 inch wide 2 inch deep board. Treated lumber is good, if you plan on keeping the board long term. But for one season, any untreated 2x8x6 (or longer) will work. I have, in the past, used untreated lumber outside and used it for several years. But treated lumber will last virtually forever.
3. Put boards (shorter ones), under your stabilizer jacks and lower them tight, but they are not designed to lift the camper, just stabilize it. Also put lumber under the tongue jack. After setting up the camper for winter storage, you should return after a few days and re-snug the jacks again. Sitting on dirt, they will sink into the ground.
4. Remove anything that may freeze and break. Anything with liquid, remove it.
5. Remove ALL food items, even stuff in cans. Water in cans can break when they freeze. It's better you bring all food items back into your house and use it up, so next Spring you start out all fresh again with new food items and nothing has hit expiration dates.
6. Depending upon where you plan to store the camper, remove all personal items that are important to you, and remove all electronics. (potential theft). If it's important to you, get it out of the camper and safeguard it at home.
7. If you are truly storing and have no plans on using it over the winter months, remove the battery and bring it home. Put it on a trickle charger over the winter.
8. Remove your propane tanks. Yes, propane tanks get stolen too.
9. Learn to winterize the camper yourself and save yourself a LOT of money, and have peace of mind it's done right. It take only 15 minutes or less to actually winterize only cost about $5 - $7 for the "pink stuff" RV antifreeze (for 2 gallons), and if you use the compressor blow-out method, actually, that costs nothing after purchasing the blow-out plug to use with your air compressor.
10. If you have an RV cover, cover it and snug it down real good. Do not use a tarp as a cover. It will cause more problems than you want if you use a tarp, rubbing against the finish and roof of your RV. If you cover, get a cover that's made for RV's and tie it down tight so it won't flop and rub in the wind.
11. Rodent protection. Do not use any kind of poisons. If a mouse come into the camper and eats the poison, then crawls into your walls and die, you will FOREVER have stink in the camper and it will NEVER go away, no matter how hard you try, unless you can break through the wall and remove the dead thing. Dead mice stink FOREVER!
12. Remember, anything you use for bait to catch critters, will also attract critters. It's better if you thoroughly inspect your camper and plug any hole or gaps with foamy spray insulation, like "Stuff" with a combination of wire mash filling the holes and gaps first. Plug any hole and the chances are critter invasion are greatly reduced.
13. If in storage, slides in! No reason to keep them out making a wider foot print and exposing the roof of the slides to bird do and snow and ice!
14. You might go ahead and over-inflate your tires by 10 or 15 pounds while it sits over the winter. Chances are, sitting, the tires will loose some air pressure and the last thing you want is to have a low or flat tire in the Spring. Just remember, in the Spring, before moving the trailer, if they are still over inflated, to bring them back to normal pressure before hitting the road. I've done this and it just gives me peace of mind.
15. When in storage, check the camper often over the winter. Don't assume, never ever ever assume someone else is watching out for your camper, even if you place it on close friends or a relative's property for storage. Check it often yourself.
Lots of tips. Hope this helps.