How to identify the aromatic compound with the sum formula C7H6Cl2?
Hints to find the structure of (A):
(1) The given compound (A), is aromatic, hence, it is highly likely to have only one benzene ring (deduced from it's index of hydrogen deficiency/degree of unsaturation)
(2) Only those halides which can give a stable carbocation at the carbon to which the halogen atom is attached, can give precipitate of AgCl with AgNO3 solution. Therefore, one of the chlorine atoms is benzylic, as you mentioned.
(3) Now, oxidation of (A) yields a monochloro-benzoic acid, which on nitration, yields only one mono-nitro derivative. Out of the possible three options: ortho, meta, and para. now, chlorine atom on a benzene ring is ortho-para directing, while the COOH group is meta directing. Therefore, if the initial compound (A) would have -CH2Cl situated para to the Cl, it's obvious that, on oxidation, the COOH group would be formed from the CH2Cl group. Now, there is only two symmetric positions, at which both COOH and Cl will direct NO2+, hence, as both are symmetrical, they can be considered to be the same molecule. For that very reason, it only makes sense for (A) to be para substituted.