The slow response is what I'm looking to factor in. As of right now the load will go from 0mA to 250mA in a few milliseconds. So I'm A) worried about the voltage droop due to the sudden load; B) calculating what size capacitor(s) can be installed to optimize the voltage droop;
For such a load change use a .1 Ufd, ceramic cap like shown in the data sheet. Place it close to the regulator output.
how well does a 7812 run with no load?
It does fine with no load. Unlike the adjustable regulators that need some external load the fixed regulators have that minimum load internally.
PS The 1000 Ufd. filter cap is kind of marginal for a 1/2 wave rectifier. I would go bigger.