Line bundles and vector bundles on $\mathbb P^1 \times \mathbb P^1$ 
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*Is there a classification of vector bundles on $\mathbb P^1 \times \mathbb P^1$?  I know that the analogue of Grothendieck's splitting theorem is not true for $\mathbb P^1 \times \mathbb P^1$.

*Is it true that any line bundle on $\mathbb P^1 \times \mathbb P^1$ is of the form $\mathcal O(m,n) = p_1^*(\mathcal O(m)) \otimes p_2^*(\mathcal O(n))$, where $p_1$ and $p_2$ denote the projections on the two factors?
 A: As Will Sawin remarks in the comments, the answer to your question (2) is "yes," either via the exponential exact sequence or by an easy application of the "theorem of the square," (which works e.g. over a general field).
The answer to your question (1) is rather more interesting--while there is no classification of vector bundles on $\mathbb{P}^1\times \mathbb{P}^1$ (that I know of), there is a splitting theorem analogous to Grothendieck's result.  First, it's worth noting that there is a result of Horrocks generalizing Grothendieck's theorem to projective spaces of arbitrary dimension:

Theorem.  (Horrocks)  Let $E$ be a vector bundle on $\mathbb{P}^n$.  Then $E$ is a direct sum of line bundles if and only if $H^i(\mathbb{P}^n, E(r))=0$ for $0<i<n$ and all $r$.

Note that if $n=1$, this is precisely Grothendieck's splitting theorem--a good place to learn about this result and its cousins is Vector Bundles on Complex Projective Spaces by Okonek et al.  It is this theorem which generalizes to $\mathbb{P}^1\times \mathbb{P}^1$.  
Indeed, a literally identical theorem to Horrocks' result above works for a quadric surface $\mathbb{P}^1\times \mathbb{P}^1$ in $\mathbb{P}^3$ (a vector bundle splits if and only if the middle cohomology of all of its twists $E(n)$ vanishes).  One reference is this paper by Buchweitz, Greuel, and Schreyer; the relevant comments are Conjecture B and Remark 2 on page 169.  Conjecture B gives a (conjectural) generalization to smooth hypersurfaces; I don't know too much about this subject, so I have no idea what the status of this conjecture is. 
Of course, this is far from a classification of vector bundles on $\mathbb{P}^1\times \mathbb{P}^1$--that said, I think that such a classification (at anywhere near the level of completeness of the classification on $\mathbb{P}^1$) is well beyond current technology.  Vector bundles on $\mathbb{P}^2$ are already quite interesting and complicated.
