Let me just expand my comment. You are talking about the uniform measure on the unit sphere of the euclidean space $Sym_n(\mathbb R)$, but for measuring subsets that are homogeneous it is equivalent to talk about the standard gaussian measure on $Sym_n(\mathbb R)$. This measure is called in random matrix theory the Gaussian Orthogonal Ensemble (GOE). In particular $p_n$ is the probability that a matrix in the GOE is positive definite. Since there are explicit formulas for the probability distribution of the eigenvalues of a GOE matrix (this is probably what Robert Bryant is proving), there migth be explicit formulas for $p_n$. 

Anyway, the asymptotics are known from general large deviation results for random matrices (due to Ben Arous and Guionnet, PTRF 1997)~: $p_n$ goes to zero as $e^{-c n^2}$ for some constant $c>0$. The constant is equal to the infimum, over all probability measures $\mu$ on $\mathbb R^+$, of the quantity 
$$ \frac{1}{2} (\int x^2 d\mu(x) - \Sigma(\mu)) - \frac 3 8 - \frac 1 4 \log 2$$
where $\Sigma(\mu)$ is Voiculescu's free entropy $\iint \log|x-y| d\mu(x) d\mu(y)$. You can probably explicitely compute $c$. It is even possible that this was known before Ben Arous and Guionnet's work, since their results are much more general.

For your second question, I am pretty sure that the limiting graph of $t \in [0,1( \mapsto p_{n,E(tn)}$ is $0$ ($E(x)$ is the integer part of $x$). But this is probably not what you really want to ask.