I am looking for a program that can spell-check a PDF document (without having to convert the PDF into another format like .docx, which I can do using Adobe Acrobate Pro). If possible, free, compatible with Windows 7 (otherwise Linux) and also checks for the grammar.
Spell check menu
Highlighted words that are misspelled
Spell check zones
For a free, cross platform, application you can use LibreOffice Writer - it will open pdf documents read and write them, (other than password protected documents that you don't have the password and documents from some scanners which are actually image collections), you can spell & grammar check in a wide range of languages.
Screenshot of checking a pdf a fiend was sent: