Option A.
We have $10$ elements for the domain set to map to. If we want to ensure $x \neq f(x)$ lets do as follows:
$f(0) = 1, f(1) = 2, f(2) = 3, \ldots, f(8) = 9.$ Now the only option is for $f(9) = 9$ or else we will break the non-decreasing condition.
Like this for any mapping we will get at least one $x$ such that $x = f(x).$
Now consider $f(0) = 0, f(1) = 0, f(2) = 0, \ldots ,f(8) = 0, f(9) = 1.$ This implies, $g(9) = g(8) = g(7) = \ldots =g(1) = 0, g(0) = 1.$ That is we have a non-decreasing $f$ and for no $x,$ $x = g(x).$ So, (ii) is FALSE.
Let $f(0) = f(1) = f(2) = \ldots =f(8) = 0,f(9) = 1.$ Now, for no $x,$ $x = f(x) + g(x) \mod 10.$ So, (iii) is FALSE.