An unmaintained beaver pond will quickly dry up as it will not hold water. The industious rodents constantly pack the inner dam face with mud and do lots of little repairs to keep it full. So you've go the right idea - Don't want a beaver pond on the farm, eradicate that particular family of beaver, the pond will dry up on its own and can be bulldozed.
Pelt care -
Skin the beaver using a cut from tail to chin on the belly side. Cut around the wrists/ankles and peel the skin (gotta cut it away, it's attatched well all around) off the carcass without any other cuts through the hide. You'll end up with a roughly circular hide.
There are several methods to stretch a beaver pelt - the easiest and my favorite is to nail it to a large peice of plywood. Start with one in nail through the hide where the nose is and then one in the tail, One to each side and then start splitting the distance between each nail until you have a circle of nails about two or three inches apart. If the hide is loose, pull the nails up and renail them after a good tug. The thing should be pretty taunt. In the end it should be pretty solid as it will try to shrink as it dries and will pull loose nails right out.
Next close off the leg holes - just stretch them out toward the outer edge and nail off with three or four nails.
Fleshing - Tough to do, so practice alot. Take a SHARP knife with a curved blade and draw it toward you, across the hide, to scrape away the fat and membrane attached to the skin. Do a good job as you'll start realizing now just how full of fat these guys are! You've got to get as much off as possible for the best tanning job. If you accidently cut a hole in the hide or find bite marks bullet holes, etc. at this point, nail them closed. (Alternate methods sew them shut with needle and thread)
About now you'll become enamored to the odor of beaver and with all that fat and grease getting into your hands and under your fingernails, you'll start to see just what it was the mountainmen saw in the trapping trade.... Or not.
Dry the hide for processing or salt treat, or start the tanning process right away. I've always sold my beaver pelts, so I won't get into the tanning process at all, - anyone else?