How do researchers ask others for comments and advices for their research?
A friend of mine wrote an undergraduate thesis that was groundbreaking original research.  Nobody at his university had the expertise to evaluate it, but one of the faculty suggested he send it to a fairly famous professor at another university.  He did.

That professor read his work and sent him some feedback, including several (correctable) errors he noticed.  He also made sure my friend's application to his program was approved and he was offered their best fellowship.  Eventually my friend attended that doctoral program and the professor he had contacted was his advisor.  He also published the a journal article based on his undergrad thesis in one of the leading journals of his field.

This is an unusual story; most "self-funded" research is not of that quality and many famous researchers don't bother to read that sort of email.  If your work is of that kind of quality, and if you are lucky enough to find a researcher who opens and reads it, this approach can work.  If you believe you're in that situation, definitely do send the manuscript in your initial email (or post it online and provide a link); the likelihood that the researcher will follow up and ask you for a copy, when he/she has no evidence of the quality or relevance of your work, is extremely low.

Always start with a salutation; the last thing you want is to come across as being rude.  I would keep the email short, but long enough to show the researcher that you've done serious research and that it's relevant to his/her interests.  Roughly the length of an abstract, though in your email you can describe things less formally and point out specific connections with the recipient's work.