Do you have propane tanks? Are they full and connected? Are there any leaks?
You'll need propane to test the stove, oven, furnace, water heater, and fridge. You'll also need a 12 volt
supply to test the furnace, water heater, fridge, and vent fans (assuming that both the fridge and water heater have circuit boards).
Do you still have the 12 volt
fuse panel? I hope so - that will make testing easier.
Do you know where the old battery used to be connected? If yes, then buy a 30 amp in-line fuse and connect the battery to the wiring harness where the old battery used to be using the in-line fuse on the positive wire as an extra safety measure. Connect the positive terminal on the battery to the fuse, and then connect the other end of the fuse to the positive wire that feeds the fuse panel. Connect the negative terminal on the battery to the white wire where the old battery used to be connected.
If you do not have the fuse panel, but a bunch of wires, then I would wait until you start pulling stuff apart to test things. Then you'll know exactly what wire goes where. You can hook up the battery to each item, one at a time, and test it to see if it works.
Depending on the model of your fridge, it may ran on 110 VAC as well as propane. Easy to tell from the front - look for a switch that says AC and Gas, or AC and Auto. If there is no switch like that, then you may have a fridge that only runs on propane, and it may not even require 12 VDC. Pictures would help.
On the roof a/c, is the original shore power cord and breaker box all still intact? If yes, then you should be able to plug the trailer into shore power and test the a/c and the fridge on AC as well. And your 110 VAC outlets. But be careful here - if the power cord, breaker box, or any of the 110 VAC wiring looks like it's been messed with by the PO, best to NOT plug the trailer in until you're sure it's safe.
To test the a/c, be sure to plug the shore power cord into at least a 20 amp outlet. Preferably not a GFCI protected outlet. A 30 amp RV outlet would be ideal, but I'm guessing you probably don't have one of those handy... To use a 20 amp outlet, you'll need a 30 amp to 20 amp adapter for the shore power cord.
The safest things to test yourself are the 12 VDC items. Especially if you and inline fuse on the battery. Worse case there is you blow the fuse if something is shorted out.
If the 110 VAC was messed with, you run the risk of getting shocked when you plug the trailer in. Again, if you're not sure, have someone that knows RV electrical systems check it out.
The propane system is another risk if you're not sure it doesn't leak. In your age trailer, both propane and water use copper lines. Propane should be flexible copper, while the water lines should be hard copper. But, if the PO messed with both systems, you may have propane lakes or even cut propane lines. Propane lines run under the trailer (below the belly skin). from the main line, there will be tees to the branch line, and maybe even shut-off valves. From the tee or shutoff valve, the propane flexible copper line runs up through the belly pan and floor to the appliance. There should not be any connections in the propane line once it enters the belly pan until it reaches the appliance.
Not trying to scare you - just making you aware of the risks involved. Pictures would help us help you figure out if something is safe or not.