What is the correct syntax to put a PDF form file name into the Subject line of a mail.Doc Javascript?

Unfortunately, there is only so much you can google to get a working solution, I would suggest that you get at least somewhat familiar with JavaScript (there are a lot of good tutorials and books available). If you want an introduction into Acrobat's JavaScript, take a look here: Beginning JavaScript for Adobe Acrobat

There is a property of the document object that returns the filename. You can read about the Doc.documentFileName property here: Acrobat DC SDK Documentation

In the most simple implementation, you can use this to get the filename into the subject line:

this.mailDoc({bUI: true, cTo: "sgeist@mailboxmerchants.com", cSubject: this.documentFileName,  cMsg: "Attached is the signed proof. Please forward to the graphics dept"});

If you want to add more information than just the filename, you will have to assemble your subject line by using string constants and variable elements. Something like this will work:

this.mailDoc({bUI: true, cTo: "sgeist@mailboxmerchants.com", cSubject: "The filename is " + this.documentFileName,  cMsg: "Attached is the signed proof. Please forward to the graphics dept"});