Two identical current-carrying co-axial loops, carry current ‘i’ in opposite sense. A simple Amperian loop passes through both of them once. Calling the loop as C.

Consider a simple Amperian loop passing once through both the identical current-carrying coaxial loops.
(i) According to Ampere’s circuital law.
$\underset{C}{\oint }\overrightarrow{B}·\overrightarrow{dl}={\mu }_0\left(i-i\right)=0$  

Option (A) is wrong.
(ii) As $\underset{C}{\oint }\overrightarrow{B}·\overrightarrow{dl}=0$, therefore $\underset{C}{\oint }\overrightarrow{B}·\overrightarrow{dl}$  is independent of sense of C

Option (B) is correct.
(iii) Depending on the shape of loop, there may be a point on C, where $\overrightarrow{B}$ and $\overrightarrow{dl}$ are perpendicular

Option (C) is correct.

(iv) $\overrightarrow{B}$ does not vanish, in general.

Option (D) is wrong.