Water testing...just do what's called a Basic Water Well test. You need to know pH, Total Alkalinity, Hardness and what basic metals and minerals exist. This test tells you whether or not to add aglime, or leave it alone, plus you get baseline info for later.
Ric's right...don't cut trees larger than 4" in diameter. Larger trees can be trimmed, but don't cut them.
Removing water...siphoning may be an option, but pumping may be the best bet without cutting the dam or reworking your stop log device.
Willows are best eradicated while growing.
Beavers...diligent trapping.
Preventing beavers...more diligent trapping. Beavers cut and build to create a lodge to live and reproduce. They will travel long distances to cut wood. So, always be on the lookout for beaver sign, small twigs stripped of bark, cut marks around trees, etc.
Woodlot planting...see above answer.
Probably no history on dam construction.

Teach a man to grow fish...
He can teach to catch fish...