Every localization is flat, and localizing a flat module preserves flatness

For an exact sequence $A\to B\to C$ of $R$-modules, [L2] gives an exact sequence $S^{-1}A\to S^{-1}B\to S^{-1}C\mathrm{.}$ By [L3] this is $(S^{-1}R{\otimes }_{R}A)\to (S^{-1}R{\otimes }_{R}B)\to (S^{-1}R{\otimes }_{R}C),$ so $S^{-1}R$ is flat over $R$ by [L1].

Let $M$ be an $S^{-1}R$-module. If $M$ is flat over $S^{-1}R$, then for any exact sequence over $R$ we first tensor with $S^{-1}R$ as in step 1.1 and then tensor over $S^{-1}R$ with $M$; the result is exact, so $M$ is flat over $R$. Conversely, if $M$ is flat over $R$, then every exact sequence of $S^{-1}R$-modules is in particular exact over $R$, and tensoring it with $M$ over $R$ agrees with tensoring over $S^{-1}R$ because the scalars already act through the localization. Hence $M$ is flat over $S^{-1}R$.

If $N$ is flat over $R$, then $S^{-1}N\cong S^{-1}R{\otimes }_{R}N$ by [L3]. Applying step 2.1 to the $S^{-1}R$-module $S^{-1}N$ proves it is flat over $S^{-1}R$.

Step 2.1 applies in particular to localization at a prime ideal by [L4].

Therefore all three claims hold.