Solution to second question in particular case, when $R_0/pR_0$ without zero divisors:

Note that $R_0/pR_0$ is finite dimensional vector space, $\dim R_0/pR_0\leq rk(R_0)$. So easy to see that $R_0/pR_0$ is field. Let for some $x\in R\setminus pR, y\in R, x^2 = py$, let $i\geq0 : r:=p^ix\in R_0, p^{i-1}x\notin R$. From defenition of $r, r\in R_0\setminus pR_0$, $R_0/pR_0$ is field, so $\exists r'\in R_0 : rr'- 1\in pR_0$. So $xrr' = p^ix^2r' = p^{i+1}yr'\in pR$, $x = xrr' - (rr' - 1)x\in pR$, but $x\notin pR$. done