Mole- I have the same problem in parts of my timber. I have mowed trails, which both I and the deer use. Makes it nice and quiet to get to your stands and is a good way to get an ATV in when you need to get a deer out.
As for killing it I'm not sure what product you are looking for. I have used plain old 2-4D, 2-3 oz to the gallon and have had pretty good luck. I sprayed in mid summer, dont know what time is best. It killed some really large plants, some where as large as a truck. I'm curious to see if they resprout this year, but it sure seemed to kill them last year. Its fairly cheap to, I used the Farm Service generic, was about $40 for 2 gallons I think. Your father-in-law probably has some already. I also like that it dosent kill the grass around the MFR. Round-up has worked too, but leaves a large dead spot.
Don't know what it would do to young oaks, but I'm guessing it would kill them to. I spot spray 2-4D carefully to avoid select plants.
I only sprayed the plants that kept growing into my trails as I did not want to thin out the timber any. I also sprayed the first 30 yards or so right behind the house so I can see into the timber a bit.