i ve had 3 refrigerator pumps, all of them run constantly without any rewiring from me. only the smallest one has some sort of thermal relay, and if you pull it out of the plug, you have to wait about 30 secs to 1 minute before it will start again.
fridge pumps have a special oil for lubrication, which is resistant to low temps, but since we arent using them for that anymore, any oil will do (i put some spare GReddy oils in mine lol)
these pumps have a suck line, and a vomit line, which expells the air they suck. after a while, oil may be spat out of there, in which case you can either put a cup there and collect it, and later feed it to the suck line, or put new oil (make it suck it lol)sluuurrp.