For every vector $α$ in $V$, the reflection associated to $α$ is the map
$$
s_α
\colon
V \to V \,,
\quad
x \mapsto x - 2 \frac{(x, α)}{(α, α)} α
$$
There are two special cases of this general formula that one should be aware off here:
If the vector $α$ is of the form $α = e_i - e_j$ for two standard basis vectors $e_i$ and $e_j$ with $i ≠ j$, then the reflection $s_α$ swaps the two standard basis vectors $e_i$ and $e_j$, and leaves all other standard basis vectors untouched.
In a formula, this reads as
$$
s_{e_i - e_j}(e_k)
=
\begin{cases}
e_j & \text{if $k = i$,} \\
e_i & \text{if $k = j$,} \\
e_k & \text{otherwise.} \\
\end{cases}
$$
If the vector $α$ is of the form $e_i + e_j$ for two standard basis vectors $e_i$ and $e_j$ with $i ≠ j$, then the reflection $s_α$ swaps the sign of both $e_i$ and $e_j$, and leaves all other standard bases vectors untouched.
As a formula,
$$
s_{e_i + e_j}(e_k)
=
\begin{cases}
-e_i & \text{if $k = i$,} \\
-e_j & \text{if $k = j$,} \\
\phantom{-}e_k & \text{otherwise.} \\
\end{cases}
$$
(One should also be aware of the case $α = e_i$, but we won’t need this here.)
The group of type $D_n$ is the Weyl group of the root system of type $D_n$. It is therefore the subgroup of $\operatorname{GL}(ℝ^n)$ generated by all reflections
$$
s_α
\quad\text{with}\quad
α = \pm e_i \pm e_j \,,
i ≠ j \,.
$$
We always have $s_{-α} = s_α$, so it suffices to consider the reflections
$$
s_α
\quad\text{with}\quad
α = e_i \pm e_j \,,
i ≠ j \,.
$$
We see that these generators of $D_n$ fall into the two categories of reflections discussed above:
The reflections of the first kind (coming from $α = e_i - e_j$) swap the standard basis vectors $e_1, \dotsc, e_n$ around.
They generated a subgroup of $W$ isomorphic to the symmetric group $S_n$
The reflections of the second kind (coming from $α = e_i + e_j$) swap the signs of the standard basis vectors. However, each such reflection always swaps two signs at once!
So if we consider the subgroup of $W$ generated by the reflections of the second kind, then we can only ever swap an even number of signs.
(This subgroup of $W$ is isomorphic to the subgroup of $(ℤ / 2ℤ)^n$ whose elements have an even number of non-zero entries. This subgroup is in turn isomorphic to $(ℤ / 2ℤ)^{n - 1}$.)
Regarding the first part of the question:
all roots have the same length, and the root system $D_n$ is irreducible (because $n ≥ 3$).
Therefore, the Weyl group acts transitively on $D_n$:
the orbit of any root is the entire root system.
We can also see this from the explicit description of the group $D_n$ above:
we can go from any root $± e_i ± e_j$ with $i ≠ j$ to any other root $± e_k ± e_l$ with $k ≠ l$ by changes indices and by swapping an even number of signs of all standard basis vectors $e_1, \dotsc, e_n$.
(We need here that $n ≥ 3$.
This allows us, for example, to go from $e_1 - e_2$ to $e_1 + e_2$ by applying the reflection $s_{e_2 + e_3}$ that flips the signs of $e_2$ and of $e_3$.
We need here the additional third standard basis vector $e_3$ to flip only a single sign in the given expression $e_1 - e_2$, but flip overall an even number of signs in all of $e_1, \dotsc, e_n$.)
