Why can we assume that squaring makes things smaller in numerical methods and Taylor expansion? If we square a number <1 it becomes smaller, but a number >1 becomes larger. 0.01 second is 100 milliseconds. 0.01^2=0.0001 (second^2) and 100^2=10000 (millisecond^2). So, the same quantity can become both smaller and larger after squaring.
A lot of things depend on squaring making things smaller: Taylor series, numerical differentiation, and integration.
How is it possible, that we seemingly can always represent the same quantity as something >1 by just changing its units of measure, and then squaring will make things larger.
How to rigorously prove that it doesn't matter for those cases when something is proportional to something squared?
 A: Nothing is large or small in itself, only in comparison to other quantities. When we say “large” or “small” without specifying a comparison, we usually mean compared to $1$, but that makes no sense for a quantity with a unit.
To explain in more detail for one of your examples, Taylor expansion: In
$$
f(x)=f(x_0)+(x-x_0)f'(x_0)+\frac12(x-x_0)^2f''(\xi)
$$
(with $\xi$ between $x_0$ and $x$), the linear approximation $f(x)\approx f(x_0)+(x-x_0)f'(x_0)$ is good if the quadratic error term is “small”. By this we might mean that it’s small compared to $f(x_0)$. The condition
$$(x-x_0)^2f''(\xi)\ll f(x_0)$$
(where $\ll$ means “small compared to”) is equivalent to
$$x-x_0\ll\sqrt{\frac{f(x_0)}{f''(\xi)}}\;.$$
Or we might mean that the quadratic term is small compared to the linear term. The condition
$$
(x-x_0)^2f''(\xi)\ll(x-x_0)f'(x_0)
$$
is equivalent to
$$x-x_0\ll\frac{f'(x_0)}{f''(\xi)}\;.$$
If you measure $x-x_0$ in other units and thus, say, multiply its numerical value by $10^3$, the numerical values of $f'$ and $f''$ are correspondingly divided by $10^3$ and $10^6$, respectively, and the smallness relations remain invariant.
So it’s not that squaring things makes them smaller; it’s that a higher power of a quantity decreases faster with the quantity than a lower power, so if you make the quantity small enough, any multiple of the higher power will at some point be small compared to a multiple of the lower power. At what point that occurs depends on the factors of the multiples, and if you change the units, the numerical values of those factors change along with the numerical values of the quantities.
