My dad has to have a kidney removed. How long will he need to use the dialysis machine after the operation?

?? This needs to be discussed with your doctor/surgeon. Why is he having kidney surgery? He may not need dialysis at all. It depends on his health problems. I can't answer this without a lot more information.
Depends. It depends on function of other kidney. If other one is working he will not need dialysis.
Depends. If your father had only one kidney removed, then he likely doesn't need dialysis. He would have one remaining kidney and that kidney would be sufficient to perform all the medical duties of two kidneys. However if that one kidney has damage to it or isn't functioning adequately after the surgery, then he may need dialysis until that one kidney recovers.