I asked Kent "Superman" Blanton, who is THE gut string guy among people I know, and he strongly advised me NOT to oil wound gut because if the core expands it can blow out the winding and potentially ruin the string.
Plain gut is another story. Dryness and acidity are the enemy. You want plain gut to be translucent; if it is not, the string is too dry and needs to be moisturized. I use olive oil with good results, others use Sno-Seal, almond oil, bore oil and a variety of other products.
When plain gut strings fray, the hairs should be dealt with before they begin to travel, which can mean the death of the string. Clip the hairs with a small scissor or nail file, or burn them off with a butane lighter (which smells AWFUL). Lightly sand the affected area with fine-grained sandpaper until it is smooth to the touch, taking care to avoid creating a flat spot, and give it a little oil.
And if you think your strings are sticky and gross now, wait for the summer humidity.