Confused about the $\pm$ sign?

If the $\pm$ sign is confusing you, get rid of it. If you have $$15 = \pm(a+x)$$ you can turn that into two equations: $$15 = a+x\\15 = -(a+x)$$ and then deal with the two equations separately, one at a time. That is exactly the meaning of the $\pm$ sign.

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The reason you're confused is because the notation is confusing! The expression $\pm a \pm b$ is actually ambiguous: in some contexts it means four values, and in other contexts it means two. Sometimes there is a convention that the two $\pm$ signs must represent the same sign; sometimes there isn't.

In your third example, I'd suggest that you write $z = \sqrt{x^2}$ and turn the equation into $z+37 = y + 40$. Then solve as usual. When you get to the end, you have $z$ and $y$. Then you can conclude that $x$ could be either $z$ or $-z$.