In general, I use the process of canceling double negatives only to help my understanding of what is being said. I would rarely use it to actually restate someone else's words, primarily because doing so often makes a subtle and undesirable change in the meaning.
But even for that first use, an important part of analyzing is to figure out the actual intended meaning of the original, and in particular to decide if the double negative is real, or only apparent. We can use your example to illustrate:
I don't want no food
That could be read figuratively, as an emphatic and even scornful declaration that the speaker does not want any food. Something like:
I don't need any help from you or anyone else. I don't need no money; and I won't accept no sympathy. And for sure, I don't want no food.
But here, the double negative isn't really a double negative. It's only apparent. So canceling would completely invert the meaning from what was intended.
However, it could also be read literally, as in:
I'm trying to lose weight quickly, so I need to drastically reduce my calorie intake. But I have to eat something. I don't want no food.
In this case, there is a negation of a negation, and so logically they can be cancelled. However, even here you'd distort the meaning slightly. The intended meaning is probably not "I do want food" or simply "I want food" -- both of which are simply the result of canceling the double negation -- but rather, "I do want a little food".
To your second example: you are right to suspect that the structure is an issue.
You couldn't live with it if it wasn't your own decision.
Both negations are real, so in that sense we have two negatives, but one is not negating the other. So you can't simply cancel them out to produce:
You could live with it if it was your own decision.
Doing that would be committing the logical fallacy known as Denying the Antecedent
The OP suggested other examples might help. Here's one I recall from the Big Bang Theory, S5E10 (YouTube Clip). Sheldon and Amy are at a movie, discussing their relationship as boyfriend/girlfriend:
Sheldon: I believe I would like to alter the paradigm of our relationship.
Amy: I'm listening.
Sheldon: With the understanding that nothing changes, whatsoever, physical or otherwise; I would not object to us no longer characterizing you as not my girlfriend.
Amy: Interesting. Now, try it without the quadruple negative.
The piece Amy is referring to, with the negatives enumerated, is:
I would not(1) object(2) to us no longer(3) characterizing you as not(4) my girlfriend.
The fact that at first glance "object" might not be seen as a negative shows how important it is to figure out the speaker's intended meaning. (Unfortunately, oftentimes the best way to figure that out is to do some cancellation of pairs, which means you have to spot the negatives in the first place. No one said this was easy!) One way to make it a bit clearer in this example is to notice that "object" could be replaced with "not agree" (strictly "not agree" is not equivalent to "object", but for illustrative purposes here it works). So you get:
I would not(1) not(2) agree to us no longer(3) characterizing you as not(4) my girlfriend.
Also, for the purposes of analyzing the logic, we can replace "no longer" with a simple "not", giving:
I would not(1) not(2) agree to us not(3) characterizing you as not(4) my girlfriend.
Clearly, since they are adjacent, the first two instances of "not" cancel . And it's not difficult to see that the second and third instances do as well. So we end up with:
I would agree to us characterizing you as my girlfriend.
And that does indeed match the gist of what Sheldon says, in response to Amy's challenge (1:36 into the clip):
Sheldon: Amy...will you be my girlfriend.
I hope I would be wrong to assume that the above won't fail to hinder your understanding. But if in fact that assumption is wrong, then I will desist from withholding an adequate apology. 😉