Consider this: $-1 \neq 1$ but $\left( -1 \right)^2 = \left( 1 \right)^2 = 1$. So, definitely, raising powers does not preserve the $\neq$.
However, addition is just translation. If you translate two distinct point, they remain distinct (in fact, with the distance preserved). So, addition (and hence, subtraction) preserves $\neq$. Similarly, multiplying by a non - zero number is same as scaling the given number. So, distinct numbers remain distinct. Hence, multiplication by a non - zero number (and hence division) preserves $\neq$.
Finally, if you multiply anything by $0$, we get $0$. So, multiplication by $0$ definitely does not preserve $\neq$.
Coming to your last question, exponentiation and logarithms are one - to - one functions. Hence, they must preserve the inequality.
Note: I gave examples of numbers only because $f \left( a, b, c, \cdots \right)$ and $g \left( a, b, c, \cdots \right)$ are actually real numbers, if $f$ and $g$ are real functions. However, if they are complex functions, you may not be able to say the same thing about exponentiation and logarithms as above. Can you think what will happen if we have two non - equal complex numbers and we raise them to $e$ or take their logarithm?