I think you have either a corrupted PDF file, or one that was not made to the PDF specification by Quartz (Apple's built-in PDF knock-off).
I couldn't change any form fields or add new ones, as you described. But using the right-click/Rename utility, got this error message which means Acrobat's form utility hit some glitch in your file.

Looking at your FIle / Properties, it shows that an out-of-date version of Acrobat Pro was used (XI), and that the "PDF Producer" (the utility that actually converts the source file into the PDF) is Apple's Quartz, the free PDF producer that comes with Apple OS. We've had PDF-compatibility problems with Quartz-built files in the past, and I wouldn't be surprised if it's at the root of your problem.

Can you re-export the PDF using an Adobe product? Then you can copy/paste the fields from the old file into the new, rather than having to recreate everything from scratch.