Learn English – Is the phrase “there’s no turning back for sure” correct

In your first example, you could very easily replace "Sure" with "Yes "and have very similar meanings. All it's trying to convey is an affirmative response, and I would not suggest that it carries and implied Please or Thank you. Think of it as short for the following:

Would you like a cup of water? - I surely would.

In your second example, the sure is definitely not trying to imply that the appreciation is expected or required. Rather it is one of several words/phrases that have come to be commonly used in reply to thanks. E.g. no problem, certainly, etc. These all have a similar usage in practice to You're welcome.

Finally, while this usage of sure is informal, it's common enough that I'd feel comfortable using it all but the most formal scenarios, with the caveat that in situations like your first example, I'd probably make it Sure, thanks since I'm being offered something.

Functionally, it performs the same role in discourse as the phrase "if you don't mind": The point of it is to indicate that the person speaking isn't ill-disposed toward the person being addressed and doesn't wish to come across as dictatorial, and yet is serious about having that person cooperate with the requested action.

The problem is that, without the softening phrase, the framing of the requested action sounds very much like a command. So the speaker is trying to maintain a delicate balance between amicability and imperative voice—at least in the OP's example ("Cut out cheap wit and listen to me, there's a good fellow") and in WS2's example ("Stop swinging from the chandelier, there's a good fellow").

I share WS2's interpretation of the words "there's a good fellow" under these circumstances as being something along the lines of "By performing the action I request, you will be acting the part of a good fellow, and will qualify for pointing out as such."