Welcome to the forums!
No, the pump would not drain the batteries, unless it was running. Do you have any water leaks, or did you drain the fresh water system before you left the trailer? If the system were drained, and the pump left on, it'll run dry until it fails or runs the batteries down.
What mode did you leave the fridge in? If you have a 3 way fridge and you set it to 12 VDC (not propane), then yes, it will run the batteries down in 24 hours or less.
Other things to consider are the size of the batteries, their charge level, etc. All will impact how long they can last.
By a drop cord, do you mean an extension cord running from a 15 or 20 amp outlet in your house? If that's what you mean, then yes, you can do that. Just don't try to run the a/c. But, it'll run the converter/charger, fridge, lights, radio, etc.