a. $\{a^mb^nc^n \mid m!=n\}$
We need to ensure #a $\neq$ #b, but also #b = #c. Both of these conditions are not restricted to any finite value and hence cannot be checked simultaneously using a PDA. So, $L$ is a CSL.
b. $\{a^mb^nc^k \mid \text{if }(m=n)\text{ then }(n\neq k)\}$
We have 2 condition to be checked here - either $m\neq n$ or $n \neq k$ in either case we accept a string and reject otherwise. Both of these checks are again not restricted to any finite values but we need not check them together as the conditions are separated by "OR". Hence this can be done using an NPDA making this language a CFL but not DCFL.
c. $\{a^mb^nc^k\mid m>n\text{ or }n<k\}$
Similar to 'b', is CFL but not DCFL.