a) The sum of the deleted numbers is always composite. Let the chosen rows be $i_1, \dots, i_p$, $p \ge 2$ and the chosen columns $j_1, \dots, j_q$, $q \ge 2$. The deleted numbers in row $i_1$ are $i_1j_1, i_1j_2, \dots, i_1j_q$.
Likewise the deleted numbers in row $i_2$ are $i_2j_1, i_2j_2, \dots, i_2j_q$ and so on; for row $i_p$ they are $i_pj_1, i_pj_2, \dots, i_pj_q$. The total deleted sum is therefore $S = (i_1 + \dots + i_p)(j_1 + \dots + j_q)$. Both factors are at least $2$ (as $p \ge 2, q \ge 2$), hence $S$ is composite.

b) The sum of all undeleted numbers can be a prime. Let the chosen rows be $2, 3, \dots, 16$ and the chosen columns $2, 3, \dots, 16$. Then the undeleted numbers are in the union of row $1$ and column $1$.
Their sum is $2(1 + 2 + \dots + 16) - 1 = 271$, which is a prime.