Personally I am not a fan of trapping. I would rather make the habitat less than appealing to them or invite some mink over. With that many ponds, you would have to do a LOT of trapping. Otherwise set up the pond banks to be mow-able, and then construct the ponds to have a fairly steep slope down into the water to discourage much plant growth.

Do some searching here (I am at work) on water tests, there are some labs for it that you can mail the sample to. I don't think aquarium or pool tests are sensitive enough to measure the lower levels of nutrients in pond water vs. aquarium water.

The "manure" is likely FA that attached to the bottom, then filled up with gas lifting the algae and some bottom sludge.

The good news is you came to the right place to get answers, and I don't think you will have much trouble getting the pond cleared up.