Example of how the multiplicity can be greater than $1$: Let $mathbb{G}_m$ act on $mathbb{A}^2$ by $t: (x,y) mapsto (t^2x, t^3 y)$. Let $v$ be any element of $mathbb{A}^2$ not on the coordinate axes, for example, $(1,1)$. So $mathbb{G}_m$ maps to $mathbb{A}^2$ by $t mapsto (t^2, t^3)$. This extends to a map $mathbb{A}^1 to mathbb{A}^2$, given by the same formula.
I'm not sure what your favorite definition of multiplicity is. Mine is to consider the map of rings in the opposite directions: $k[x,y] to k[t]$ by $x to t^2$, $y to t^3$. The point $(0,0)$ corresponds to the ideal $langle x,y rangle$; this maps to the ideal $langle t^2, t^3 rangle = langle t^2 rangle$ in $k[t]$. Then, by definition, the multiplicity of $f^{-1}(0)$ is the dimension of $k[t]/langle t^2 rangle$, which is $2$.
The geometric point here is that the map $mathbb{A}^1 to mathbb{A}^2$ has vanishing derivative at $0$, so the preimage of zero has multiplicity greater than $1$.
You also asked about a Weyl invariant form on $mathrm{Hom}(mathbb{G}_m, T)$, where $T$ is a maximal torus of $GL_n$. Every hom from $mathbb{G}_m$ to $T$ is of the form $t mapsto (t^{w_1}, t^{w_2}, ldots, t^{w_n})$, where the coordinates on the right hand side are the entries in your diagonal matrix. Let's call this $s(w)$.
The obvious choice of Weyl invariant form is
$$langle s(v), s(w) rangle = sum v_i w_i.$$
Technically, I note that $sum v_i w_i + c (sum v_i) (sum w_i)$ would also be Weyl invariant, for any integer $c$. No one ever makes the choice of using nonzero $c$, but I don't know a general principle which would exclude it.
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