Can small truck pull a livestock trailer?

No. There's a certain weight limit that a truck can pull without putting stress on the transmission. It's usually in the owner's manual of your truck that you will find it. A trailer is a heavy thing to pull, and even heavier when you have animals in it. You're better off with a larger pick-up that has the weight specifications to be able to pull a livestock trailer. This is really important if you have large farm animals like cattle or horses that you have to truck around.