If the mouth rot is bad enough that the gecko has trouble eating it should be treated by a vet (it may require antibiotics and/or daily cleaning with nolvalsan or betadine). You also mention an underbite--are you sure your gecko doesn't have MBD? Are you supplementing with calcium and/or any other vitamins?
As for the prolapse/sperm plug, if it is a prolapse you do NOT want to pull at it at all. Prolapses are serious and require immediate veterinary attention (within 24 hours) or the animal may lose the entire hemipenis (if that is what is prolapsed). You can put a paste made from sugar and water on the prolapsed area to help draw the moisture out of it to help it to shrink before you can get the gecko to a vet. Other than that, keep it as clean and dry as possible (use paper towels for substrate).
If it is just a sperm plug it will be dry looking and not moist at all. If you can post a photo we may be able to determine what it is for sure, but until you know do not attempt to remove it.