Trenching is expense, some people get around that by using multiple
bore holes instead of trenches ... your just trying to keep the stock
tanks from freezing over .. I wonder if a single deep bore hole would do
the trick.

The best way to keep the gunk and fish out of the loop is to use a
closed loop, that way nothing can get in to the loop. Heat can
transfered between the loop and the tank with a heat exchanger.

PVC can be used but, most horzonal systems use HDPE - high density
polyethylene pipe, good for 50+ years.

As for pipe size ... there is some data collection and math to do before
that can be answered.