a$)$ In this scenario, there was a contract between Barry and Melinda. The essential elements of a contract, such as offer, acceptance, consideration, legality, capacity, and certainty, seem to be present. Barry offered the pink leather jacket for sale, Melinda accepted by purchasing it for $$400$ in cash $($consideration$),$ and both parties had the capacity to enter into the contract. Legality is presumed unless there is evidence to the contrary. Therefore, based on these elements, a valid contract was formed, and legal title to the pink jacket was transferred to Melinda.

b$)$ If Barry argues that he was mistaken as to the identity of Melinda, it might not necessarily render the contract void. Mistake as to identity typically applies when one party intends to contract with a specific person but mistakenly contracts with someone else. In this case, the mistake, if any, could be related to Melinda's false representation rather than a mistake as to her identity. Barry may have a case for fraud or misrepresentation, but it may not automatically void the contract.

c$)$ Relevant legal principles and cases that could be considered include:

i$.$ Mistake as to Identity: The case of Cundy v Lindsay $(1878)$ is often cited in situations involving mistake as to identity. However, the applicability of this case may depend on the specific details of the mistake.

ii$.$ Fraud or Misrepresentation: If Melinda intentionally deceived Barry by falsely claiming to be the Mayor's daughter, this could be considered fraud or misrepresentation. Cases like Derry v Peek $(1889)$ establish the elements of fraudulent misrepresentation.

iii. Stolen Property: If Barry believes that Valerie has stolen the jacket, this could be a separate legal issue. However, it's important to establish whether Valerie was aware that the jacket was obtained through fraudulent means.

It is recommended that Barry consults with a legal professional to discuss the specific details of the case and the applicable laws in the jurisdiction. Legal advice tailored to the specific circumstances is crucial in complex situations like these.