Script to Copy Text in Proper Case (Each Word Capitalized)

You first need to define if you want sentence case or title case. If title case, that can change based on a set of style parameters, such as Chicago Manual of Style or AP Style. 

THIS IS ALL CAPS.

This is sentence case.

This Is Tile Case in Chicago.

Title Case Won't Be the Same When Using AP Style in Terms of Which Letters Are Capitalized.

You can use this function to convert a string to this "proper" case (I know it as Title Case):

function toTitleCase(s) {
	var words = s.split(" ");
	for (var i in words)
		words[i] = words[i].charAt(0).toUpperCase() + words[i].substr(1).toLowerCase();
	return words.join(" ");
}

Note that the above can yield strange results. For example, if you run it on "I'm going to visit the USA.", it will return "I'm Going To Visit The Usa."...

As mentioned, though, copying the text to the clipboard is tricky.

Sure, if you can define "proper" case (which would not normally be "Proper Case" for English) then JavaScript can do it. HOWEVER, you can't put it back on the page to copy. Better to process it with an external app once it's on the clipboard.

The Acrobat JavaScript model does not provide direct access to the selected text or the clipboard.  However, an indirect/partial solution would be a script that runs the Copy menu item.

app.execMenuItem("Copy", this);

It's a partial solution because the context of the copy is not guarenteed to be the selected text.

I also think this might require a privileged context.

https://www.pdfscripting.com/public/Trust-and-Privilege-in-Acrobat-Scripts.cfm