My question is if there exists a way to evaluate the sum
$$ {{s}\choose{s}}^{\!2} + {{s + 1}\choose{s}}^{\!2} + \ldots {{s+r}\choose{s}}^{\!2}. $$
In other words, it's the sum of the squares of the first r binomial coefficients on the s-th right-to-left diagonal of Pascal's triangle. Moreover, is it true that the previous sum is $O_{\!s}(r^{s})$?
I don't think there's a nice closed form.
For the second question, perhaps you mean $O(r^{2s+1})$, not $O(r^s)$. In fact, for each fixed positive integer $s$, ${s+x \choose s}$ is a polynomial in $x$ of degree $s$, so ${s+x \choose s}^2$ is a polynomial in $x$ of degree $2s$, and the partial sum ${s+0 \choose s}^2 + {s+1 \choose s}^2 + \ldots {s+r \choose s}^2$ is a polynomial in $r$ of degree $2s+1$.