Term symbols for nitrogen: determining J
Firstly, regarding the extra question: Atkins' Molecular Quantum Mechanics (5th ed.) uses "term" for $^2\!S$, and "level" for $^2\!S_{1/2}$.

Back to the main question. It's been a long time since I did term symbols, so I am happy to be corrected, but if I am not wrong, your $^4\!P$, $^4\!D$, and $^2\!S$ terms are not allowed because of the Pauli exclusion principle. The argument is pretty similar to the one in the comments on my answer, which you're already aware of.

A term with $S = 3/2$ must have a set of states with $M_S = +3/2, +1/2, -1/2, -3/2$. Likewise, a $P$ term with $L = 1$ must also have a set of states with $M_L = +1, 0, -1$. Consequently, the total number of states associated with a single term $^{2S+1}\!L$ is $(2S+1)(2L+1)$. (The total number of states in a level, $^{2S+1}\!L_J$, is $2J+1$, and you can also show that the sum of this over the allowed values of $J$ is equal to $(2S+1)(2L+1)$.)

One of those twelve states in the purported $^4\!P$ term must have $(M_S, M_L) = (+3/2, +1)$. However, $M_S = +3/2$ can only be achieved if all three electrons have the same spin (spin up in this case), which in turn necessitates that the three electrons are all in different p-orbitals, such that $M_L = \sum m_l = +1 + 0 + -1 = 0$.

Ergo, a state with $(M_S, M_L) = (+3/2, +1)$ cannot exist, and the entire term $^4\!P$ cannot be possible. The same argument also applies to the $^4\!D$ term.

I suspect that showing that the $^2\!S$ term is forbidden is somewhat trickier. I had a similar issue before with carbon, which is an easier system to understand (only two electrons to consider instead of three), so maybe you will find this useful: Pauli-forbidden term symbols for atomic carbon.

Out of the three surviving terms ($^2\!P$, $^2\!D$, $^4\!S$) you already see from NIST that all the possible $J$ values (from $|L-S|$ to $L+S$) are allowed, so there is no issue with your calculation of $J$.