Converting to $\operatorname{N}(0,1)$:
a)
\[\begin{eqnarray*}P(\var{lower1} \lt X \lt \var{upper1})&=&P(X \lt \var{upper1})- P(X \lt \var{lower1})\\&=&P\left(Z \lt \frac{\var{upper1}-\var{m}}{\var{s}}\right)-P\left(Z \lt \frac{\var{lower1}-\var{m}}{\var{s}}\right)\\&=&P(Z \lt \var{zu})-P(Z \lt \var{zl})=\var{precround(u,4)}-\var{precround(l,4)},\\&=&\var{p}\end{eqnarray*}\] (to 3 decimal places.)
b)
We need to find $x$ such that:
\[\begin{eqnarray*}P(X \lt x) &\le& \var{percentile/100}\\ \Rightarrow P\left(Z \le \frac{x-\var{m}}{\var{s}}\right) &\le& \var{percentile/100}\\ \Rightarrow \frac{x-\var{m}}{\var{s}} &\le& \Phi^{-1}(\var{percentile/100})=\var{precround(normalinv(percentile/100,0,1),4)}\\ \Rightarrow x &\le&\var{precround(normalinv(percentile/100,0,1),4)}\times\var{s}+\var{m}=\var{perX}\end{eqnarray*}\] to the nearest whole number.
Hence $X=\var{perX}$ gives the $\var{percentile}$% percentile.