The coefficient 3 in the denominator of the O(h^2) term
of Q4 should be 12 and not 3 (i.e. it is 2*3! and not 3!/2).
A1 has been updated.
Also, in Q3d, I have rephrased the question
“what is the maximum number of terms … for any x” to
"what is the minimum number of terms (or what is the minimum $n$)
that, for any $x$, no matter how large,
guarantees to keep the absolute value of the remainder
$|R_{2n+2} |$ below $10^{-16}$?
We are looking for the minimum n, that, for the worst x (max x), keeps remainder below the given value.