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 $\mathbb{P}^1$.

But $\mathbb{P}(O \oplus O \oplus O(1))$ defines the blow-up of $\mathbb{P}^3$ along a line. (The proof is similar to the surface case).

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.