If you are fulltiming, I suspect you will bring much more stuff than you think. Anyway, after a few months, you will probably have even more stuff. We carry lots of stuff in the truck bed—the dirty things like boxes for sewer hoses and connections, water hoses, tools, hitch parts, chairs, table, spare tire (we have changed to 16" wheels and they don't fit well in the tire carrier), and more including a generator and gas cans.
An empty or partly empty gas can or tank is more explosive than a full one because the gas fumes are highly volatile (liquid gas is too, but less so). It can get pretty hot in a truck with any kind of cover on it and while it is very unlikely it will blow, I'd rather have an explosion in the truck bed than the trailer or truck cab. We use the truck cab back seat area to store tools and more camping stuff. But if we had a dog, the dog would go there. Would you put your kids or grandmother in the truck bed?
At RV stores, you can get foam sleeves to go over china, glasses and anything breakable. Bubble wrap works well too, but sleeves are easy. Plastic and paper dinnerware is a good choice, but a few real plates and bowls are nice to have. Wrap them and make sure everything is tightly packed to prevent them from jumping around. Put the heaviest stuff in compartments close to the floor—that lowers the trailer center of gravity. Try to put equal weight on both sides of the trailer and front to back. This makes the trailer more stable.
If you have a bedspread with some friction (not smooth) things will pretty much stay where you put them (unless you drive 60 on washboard roads). I've put computers, cameras and other valuable stuff on the center of the bed and they've never moved.
Why do you need an electric cooler? If it operates on 12 v., it may draw down the battery a bit. You have a 7 cubic foot fridge. We have travelled up to 8 weeks and the fridge was packed, but it was enough. It may be difficult to find healthy foods in some places, but we managed fine. We are very particular about what we eat, but we've survived in northern Canada, Alaska, and even Kansas. Those portable solar panels help, but we find 200 watts of solar keeps us from ever having to use the generator (unless it is late fall and we are camped under big trees in a narrow canyon like Yosemite). We hardly ever use the generator and could go without. If you don't have LED lighting, you will use much more battery and consider changing to high quality LED's. Solar has gone down a lot in price, so permanently mounted ones may be a good choice. Lewster, a forum member, is great with solar and can help you make decisions or you can swing by his shop and have them installed.
Lots of people store things on the battery box. Note than rocks and dirt hit that area and can damage things and a good cover is a good idea. It can increase tongue weight significantly though. Airstream gives tongue weight numbers that are a lot lower than reality, so add a few hundred pounds to what they say for your trailer (720 lbs. I believe) when figuring your weights for towing.
Anything on the floor will move around a lot. The smoother the surface, the more movement. That's true of things left on the seating too. After a section of bad road, the pillows that come for the seat cushions are usually on the floor.
It is a great idea to take some short trips to get used to this kind of travel. You will learn a lot and make a lot of mistakes at first. Things will break in the trailer whether it is new or used. I bring enough tools to take apart a locomotive (I'd need a new brain to put it back together), but every one of them gets used over time. You also should collect spares of things that break—like the catches that keep the drawers closed. Screws will back out and get some thread locker for them and super glue for things that don't stay glued. You'll need cleaning products—cleaner for the plastic rock guards is different than window cleaner, silicon for the rubber seals on the windows and doors, chemicals to keep the black tank smelling good, and more stuff that you can only learn from experience. Woodall's RV Owners Handbook explains how things work and how to fix them. And read everything you can on the Forum to learn.
It looks like you haven't done this before. You have a steep learning curve, but we all did and most of us have figured it out though experience and lots of reading. But some just venture out and get discouraged and wish they've never thought of fulltiming or even just traveling several weeks a year. I'd take more trips, each one longer than the last, to get used to this big change in your life. After 50,000 miles traveling and lots of reading, we still are learning and always will.
Good luck and keep asking questions.