Computing $\pi_4(S^3)$ using Serre spectral sequence

First: Are there any other terms $E_2^{p,q}$ with $p+q=5$ which survives to $E_\infty?$

No, in fact, the only non-trivial $E_2^{p,q}$ with $p+q=5$ is $E_2^{3,2}$. To see this, it may be useful to observe that $H^p(S^3;\mathbb{Z})$ is non-trivial only at $p=0,3$, and that $\mathbb{C}P^\infty$ is a model for $K(\mathbb{Z},2)$, so $E_2^{0,5}$ is also trivial. If there is only one non-trivial factor in your filtration...

Similar considerations also answers the second part of this question.

Second: You'll probably have to use the fact that $\pi_i(S^n)$ is finite for $i \not=n $, $n$ odd, which should help you calculate the Ext group. This fact follows from Serre's mod $\mathcal{C}$ theory, which you would undoubtedly have seen before if you're calculating homotopy groups of spheres.