By definition, $x\geq y$ if and only if $x-y=0$ or $x-y$ is the positive class.
By definition, $\mathrm{abs}(a) = a$ if $a$ is in the positive class or $a=0$, and $\mathrm{abs}(a)=-a$ if $a$ is not in the positive class and not equal to $0$ (that is, if $-a$ is in the positive class).
So, for example, to prove (1), if $a=0$, then the inequalities hold. If $a$ is positive, then $a+a$ is positive (because the positive class is closed under sums), so $a - (-a)$ is positive, proving that $a\geq -a=-\mathrm{abs}(a)$; proving $\mathrm{abs}(a)\geq a$ when $a$ is positive is easy. Now you can consider the case where $-a$ is positive.
For (2), if either $a$ or $b$ are $0$, the inequality is easy. If $a$ and $b$ are both positive or both negative, the inequality is again easy. If $a+b=0$ then it's easy as well. Check what happens when $a$ is positive and $-b$ is positive (there are two cases, depending on whether $a+b$ is positive, or $-(a+b)$ is positive).