It took a little while ;-) but there were some clear symptoms. And, one way to have seen the issue (though I figured it out before this, but I was going to try this next) is that, if you call the manipulate output myManipulate and then execute
InputForm[myManipulate]
you get the following result (in the original problematic case):
Manipulate[If[\[Theta]1 < \[Theta]2, GraphicsRow[{gz, gw[\[Theta]1, \[Theta]2]}, Background -> bgColor,
Spacings -> 0, ImageSize -> Large], warning],
Style["Choose 1st and last points of domain.\nTry [0, \[Pi]/2] and [\[Pi]/2, 2\[Pi]].", Bold,
16], Delimiter, {{\[Theta]1, 0, Style["First", Bold, 12]},
{0, Pi/12, Pi/6, Pi/4, Pi/3, (5*Pi)/12, Pi/2, (7*Pi)/12, (2*Pi)/3, (3*Pi)/4, (5*Pi)/6,
(11*Pi)/12, Pi, (13*Pi)/12, (7*Pi)/6, (5*Pi)/4, (4*Pi)/3, (17*Pi)/12, (3*Pi)/2,
(19*Pi)/12, (5*Pi)/3, (7*Pi)/4, (11*Pi)/6, (23*Pi)/12, 2*Pi}},
{{\[Theta]2, 2*Pi, Style["Last", Bold, 12]}, {Pi/12, Pi/6, Pi/4, Pi/3, (5*Pi)/12, Pi/2,
(7*Pi)/12, (2*Pi)/3, (3*Pi)/4, (5*Pi)/6, (11*Pi)/12, Pi, (13*Pi)/12, (7*Pi)/6,
(5*Pi)/4, (4*Pi)/3, (17*Pi)/12, (3*Pi)/2, (19*Pi)/12, (5*Pi)/3, (7*Pi)/4, (11*Pi)/6,
(23*Pi)/12, 2*Pi}}, ControlPlacement -> Top,
FrameLabel -> Style["Multifunction Domain Selection", 20, Bold], Initialization -> Null]
And there is the smoking gun in the form of Initialization -> Null.
By the way, another way to handle getting this much code into the Manipulate without having to copy it all into the Initialization option, is to have the code external to the Manipulate and then five the Manipulate the SaveDefinitions->True option. This will automatically place the needed definitions into an internal Initialization option.
Finally, you probably want to update the notebook that's on your https://faculty.unlv.edu/gbp/mma/VisualComplexAnalysis/CDF/multifunctionCDF.nb page to the updated Initialization option as a delayed rule.
I''m glad this all worked out!
--David