Don't be too hard on yourself. On a 2002 refrigerator you have no idea if the previous owner ran it off level for years and it was already failing before you even bought it.
Before running on gas for the first time make sure you light the stove burners and purge any air out of the propane lines. Otherwise it may never light and cool.
For all you know it might not even be running due to a blown fuse or air in the line and therefore isn't going to be damaged in your back yard. As others have said level the trailer, put a water bottle or small thermometer inside close the door and check it again in the morning. If it's not cold then flip the cooling method from gas to electric or vice versa and check it again in another day.
Also the checking the chimney for heat is a good idea. But don't just grab it without first hovering hour hand near it to see if it's hot. Another test is to turn the refrigerator off for about 20 - 30 minutes then go stand with your ear by the outside access panel and have a helper turn the refrigerator on to the gas setting. After a few seconds you should hear the igniter click three times and then a small roar as the flame lights. No flame, no cooling, no damage.
Good luck with it and let us know what you find.