Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?

In general, if an abelian variety $A$ contains an abelian subvariety $B\subseteq A$, then $A$ contains another abelian subvariety $B'\subseteq A$ such that $A$ is isogenous to $B\times B'$. This is Poincaré's reducibility theorem. (See also Poincaré's complete reducibility theorem, same book, next page). An abelian variety is called simple if it does not contain any nontrivial abelian subvarieties. Simon's argument shows that there exist simple complex tori of dimension 2. One could also count moduli and conclude that not every abelian surface (or abelian variety of arbitrary dimension $>1$) can be isogenous to a product. 


No. In general, there are no elliptic curves on an Abelian surface. Thinking in terms of lattices, if there is an elliptic curve on $A$, then there is a rank 2 sublattice of $\Lambda$ (Where $A = \mathbb{C}^2/\Lambda$) which is invariant under multiplication by $\sqrt{1}$. But it is easy to see that there are many rank four lattices in $\mathbb{C}^2$ for which this is not true. 


You may have a look to: Ernst Kani, Elliptic curves on Abelian surfaces (the credit for this reference goes to Dan Petersen who already suggested it in a comment to this question) 

