ていただけないでしょうか /て いただけませんか.

All are used for petition. 

ていただけないでしょうか is the politest. 

Then, ていただけませんか is a bit less polite, and くださいませんか is less polite though all are polite way of petition. 

いただくis the humble form of もらう and くださる is the honorific of くれる. So, ていただけないでしょうか /ていただけませんか is the humble way of てもらえませんか, and てくださいませんか is the honorific of くれませんか. 

I think it's very comlicated for the learners even for us and I wonder if my explanation is perfectly correct or not I hope other people, especially, professional teachers help you. 

Hey :)

These are 敬語 (Keigo)　related questions and actually very hard to master for us, foreigners. If you ask japanese people, most of them will not be able to answer that because it's commonly accepted as a natural concept for them.

The japanese honorific system is complex, as you already know so don't worry too much about it :)

Basically, what we need to know is : There are 3 ways to display respect in Japanese.

尊敬語 (Sonkeigo) When you want to show yourself polite by "elevating" the person you're referring to.

謙譲語 (Kenjougo) When you want to stay humble and "lower" yourself.

丁寧語 (Teineigo) When it's just about being polite on a general way.

I know, the meaning can "sound" the same for us, westerners but it's all in the shades.

The ていただけないでしょうか /　ていただけませんか forms are called 謙譲語　(Kenjougo). The humble and modest way to express yourself. They are both very similar but the second one sounds even slightly more formal.

You use them when you ask someone to do something for you while "lowering" yourself.

In your example :

- 申し訳ないんですが、ちょっと見ていただけないでしょうか

I would translate the idea, not word-for-word, as : I'm so sorry but, would you mind having a look to this "bad/wrong/not as good as if you would have done it" thing for me?

- 細かいのに替えてくださいませんか。 This form is called 尊敬語 (Sonkeigo). You use that form for polite requests when you are conscious about your own status and the one you're talking to :
Here, in your example above, I understand that you have a polite request to formulate to someone who is under you. Like when you are shopping, as a customer and talk to someone who works there :

It's small "coins" but, would you mind changing them for me, please?

Somehow, if you would do it for me it would be very nice (but I expect you to do so). Here, we focus on the social status and the movement. Who is talking? To who and what is moving from where to where?

Believe me, 敬語 (Keigo) is a very hard system to overcome. It can get very deep and twisted but don't be discouraged because Japanese people will be amazed if they hear you using it correctly :)

Cheers !

Alex