Looking at the roof it is hard to believe there haven't been any leaks. In a nearly 44 year old trailer, I would expect almost any exterior penatration would have leaked about once at some time.
If the leaks were bad, your Mark I nose will generally detect the mildew smell.
On the refer, usually after a couple of hours the screws on the evaporator might start developing frost. Of course I live where humidity in the 100% range is pretty much a daily occurance.
The other thing to consider is whether the trailer uses aluminum wiring. That would not be a show stopper but your maintenance and inspection will be different.
You have a hot water tank and heater. I would inspect both carefully.
On the refer, you really need and all day run to be certain. However, because the refrigerant in ammonia based, I would think if it is cooling at all, unless you detect the odor of ammonia it is probably OK. YMMV, but to really be certain I would let it run overnight just to be certain.