All of those expressions mean "won't you (please) …"
〜てくれない would be used among friends or other people who are below you in social status. 〜てもらえない is just a bit more polite than that, and still can be used around people who you're close with in the same social status. 〜てくれませんか and 〜てもらえませんか are the polite forms of the previous two, and can be used around people who you're not that close with or people you work with. 〜て下さいませんか is even more polite and can show that you're lowering yourself to the person you're talking to. 〜ていただけませんか is the most polite out of all of them and can be used when you're in a formal business setting or when you're asking something from someone who is of a higher position than you.
In Japan, social standing plays an important role in the formality of speech that you choose. For example, social standing may also be determined by who is older or younger.