For a number $n$ that has $b=1+\log_2 n=O(\ln n)$ bits
NFS has complexity of the order
$$
\exp\left\{(c+o(1)) (\ln n)^{1/3} (\ln \ln n)^{2/3}\right\}
$$

which is subexponential when compared to the input size in bits. An exponential algorithm would have
complexity of the order
$$
\exp(c b)=\exp(c \ln n).
$$

If the unusual term factorial time means an algorithm takes time asymptotic to $b!$ for the input size $b$ the answers to your questions are both yes.