What kind of problem is "email text extraction"? I need to retrieve just the text from emails. The emails can be in HTML format, and can contain huge signatures, disclaimer legalese, and broken HTML from dozens of forwards and replies. But, I only want the actual email message and not any other cruft such as the whole quotation block, signatures, etc.

This isn't really a problem that could be solved with regex because HTML mail can get very, VERY messy.

Could a neural network perform this task? What kind of problem is this? Classification? Feature selection?
It's certainly possible to treat this as a natural language processing problem, basically you're looking to assign "salience" scores to the text.

Really, though, that's overkill for this kind of problem. Writing a regex or a CFG parser (or better: finding an existing parser) is likely to be easier and more reliable.