The cone $D_a$ has a singular point of type $\frac{1}{a}(1,1)$ at its vertex. Blowing up the vertex, the exceptional divisor is a curve $C \subset Y_a$ isomorphic to $C_a$ and whose self-intersection is $\deg \mathcal{O}_{C_a}(-1)=-a$.
Since $Y_a$ is clearly a geometrically ruled surface over a rational curve (the ruling is given by the strict transform af the system of lines of $D_a$) and $C$ is a section of self-intersection $-a$, it follows $Y_a \cong X_a.$
Conversely, starting from the surface $X_a$ one can consider the unique section $C$ of negative self-intersection, namely $C^2=-a$; then this section can be blown down by Artin contractibility criterion.
The blow-down of $C$ is precisely the map $\varphi$ associated to the complete linear system $|C+aF|$ in $X_a.$
Indeed $h^0(X_a, C+aF)=a+2,$
hence $$\varphi \colon X_a \longrightarrow D_a \subset \mathbb{P}^{a+1}.$$
It is immediate to check that $\varphi$ is birational onto its image $D_a$, that it contracts $C$, that the ruling of $X_a$ is sent into a family of lines passing through the point $\varphi(C)$ and that a general hyperplane section of $D_a$ is a rational normal curve $C_a$ of degree $a$.
Therefore $D_a$ is a cone over $C_a$ and $X_a$ is isomorphic to the blow-up of $D_a$ at its vertex $\varphi(C)$.
The first thing I would like to point out is that the variety we care about is actually
{$([x_0 , x_1 ],[y_0,y_1,y_2]):{x_0}^n y_1 - {x_1}^n y_2 =0$}$ =V \subset \mathbb{P}^1 \times \mathbb{P}^2$
(the defining equations you gave are not usually homogenous, and this is what Huybrechts gives in his book: http://books.google.com/books?id=eZPCfJlHkXMC&q=hirzebruch).
We want to start out by thinking about $\mathbb{P}^1 \times \mathbb{P}^2$ as a $\mathbb{P}^2$ bundle over $\mathbb{P}^1$. In particular, it comes with the distinguished line bundle $\pi_2^*\mathscr{O}_{\mathbb{P}^2}(1)$ ($\pi_2$ is the projection onto $\mathbb{P}^2$). Now we examine the short exact sequence of sheaves of modules which defines V:
$0\rightarrow \pi_1^*\mathscr{O}_{\mathbb{P}^1}(-n) \otimes \pi_2^*\mathscr{O}_{\mathbb{P}^2}(-1)\rightarrow \mathscr{O} _{\mathbb{P}^2\times\mathbb{P}^1} \rightarrow \mathscr{O}_V\rightarrow0$.
We would like to apply ${\pi_1}_*$ on this sequence after twisting by $\pi_2^*\mathscr{O}_{\mathbb{P}^2}(1)$. The point of this is that if you believe that $V$ is a $\mathbb{P}^1$ bundle over $\mathbb{P}^1$ (which you should because you can check it on charts) then ${\pi_1}_*(\mathscr{O}_V \otimes \pi_2^*\mathscr{O}_{\mathbb{P}^2}(1))$ should be the corresponding locally free sheaf of rank 2. So we can then check to see if it is isomorphic to $\mathscr{O} _{\mathbb{P}^1}\oplus\mathscr{O} _{\mathbb{P}^1}(n)$.
After twisting and applying ${\pi_1}_*$, we get the sequence:
$0\rightarrow \mathscr{O}_{\mathbb{P}^1}(-n)\rightarrow \mathscr{O} _{\mathbb{P}^1}^{\oplus 3} \rightarrow {\pi_1}_*(\mathscr{O}_V \otimes \pi_2^*\mathscr{O}_{\mathbb{P}^2}(1))\rightarrow0$.
(In general one does not get exactness on the right, but in a situation like this we do)
And by figuring out what our maps are doing we should be able to argue that the sheaf on the right is some line bundle twist of $\mathscr{O}_ {\mathbb{P}^1}\oplus \mathscr{O}_{\mathbb{P}^1} (n)$, which implies $V \cong \mathbb{P}(\mathscr{O} _{\mathbb{P}^1}\oplus\mathscr{O} _{\mathbb{P}^1}(n))$.
Best Answer
The 3-dimensional analogues should be $\mathbb{P}(O(a)\oplus O(b)\oplus O(c))$, yes. These are exactly the $\mathbb{P}^2$-bundles over $\mathbb{P}^1$.
In general, any projectivization of a vector bundle on $\mathbb{P}^1$ is toric. (See the book by Cox, Little, Schenck). If I remember correctly, all smooth toric varieties of picard number 2 is of this form.
The blow-up of $\mathbb{P}^3$ at a point is however not of the $\mathbb{P}(O(a)\oplus O(b) \oplus O(c))$, because the blow-up does not even admit a morphism to $\mathbb{P}^1$.
However, it is a projective bundle over $\mathbb P^2$; projection from a point gives a morphism $$Bl_p\mathbb{P}^3\to \mathbb P^2$$ which is a $\mathbb {P}^1$-bundle over $\mathbb P^2$. Explicitly, it is given by $\pi:\mathbb{P}(O \oplus O(1))\to \mathbb P^2$.
Finally, $\mathbb{P}(O \oplus O \oplus O(1))$ defines the blow-up of $\mathbb{P}^3$ along a line. (Using the Hartshorne notation for $\mathbb P(\mathcal E)$).
The proofs of these statements are similar to the surface case.