With weather like we've been having, how much would an average size farm pond go down each day? Let's say the pond sits out in the open with no inflow, outflow, or trees or shade.

It's a very complicated formula based on surface area, air temp, humidity, wind speed and a couple of other variables. When I worked at the BRA we had a setup at each of the lakes to calculate it for us. IIRC it wasn't unusual for it to be an inch a day or maybe more.

Not sure either but this thread caught my eye-
I dug a pond 18 months ago and we immediately got rain and and it’s held a good water level until recently, it’s now the lowest it’s ever been-
But I had it dug to 20 foot at the deepest and really only the most shallow areas have been exposed-
I throw fish food in there every day and I still see plenty of fish activity for whatever that’s worth-
It’s still in a lot better shape than many ponds I drive by in my area though-

I was told by a well respected lake biologist, about an inch per day is a general rule of thumb in this kind of weather.

A lot of it will depend on the contour of the bottom as well. Shallow areas tend to heat up more, a body of water with a lot of good general depth will handle it better.