Let's say I have a set of pairs $(a, b)$ with real components. Is it possible to define a norm on this set that satisfies $||(a, b)|| = 0 \iff a(a+b)=0$?
Generally speaking, does a function $f(a, b)$ exist with these properties?
$$f(a,b)≥0$$ $$f(a,b)=0 \iff (a=0 ∨ a+b=0)$$ $$f(ka, kb) = kf(a, b)$$ $$f(a,b)f(c,d)=f(ac,ad+bc+bd)$$ $$f(a+c,b+d)≤f(a,b)+f(c,d)$$
The last three constraints should show how multiplication and addition on these "pairs" work.
If any such function or a family thereof exists, what is it?
No. In particular, for $a \neq 0$, we have $$f(a,-a) + f(0,a) = 0$$ since $a(a + -a) = 0(0 + a) = 0$ but $$f(a + 0, -a + a) = f(a,0) > 0$$ since $a(a + 0) = a^2 > 0$.
Thus, no function exists that satisfies your first, second, and fifth conditions simultaneously.