Equivalence Partitioning involves categorizing input data into equivalence classes. This helps in identifying edge cases by selecting representative test cases from each class, ensuring thorough coverage and testing for various scenarios. It allows testers to focus on specific data sets that are likely to expose potential issues in the API.

When an API deprecation is likely to cause disruption, it's advisable to provide a long deprecation period with alternatives and backward compatibility. This allows users to migrate at their own pace, reducing the impact on existing workflows. Abruptly shutting down the API or extending the deprecation period indefinitely can lead to frustration and business disruption.

In scenarios where API tests need to run across different platforms and devices, Parallel Execution is the most beneficial feature of an automation tool. This allows simultaneous execution of tests across multiple environments, ensuring efficient testing across various platforms and devices. It helps in identifying any platform-specific issues and improves overall test coverage.

In microservices, API versioning is crucial to ensure smooth transitions and backward compatibility. It allows existing clients to work seamlessly with updated APIs.

In scenarios where mobile applications need lightweight data exchange, RESTful APIs are preferred due to their simplicity, scalability, and ease of integration with mobile platforms. RESTful APIs use standard HTTP methods and have a stateless architecture, making them suitable for mobile applications.

Boundary Value Analysis in API testing is especially useful for uncovering security issues in input validation. By testing values at the boundaries of accepted input ranges, potential vulnerabilities related to security, such as injection attacks, can be identified and addressed. Security is a critical aspect of API testing, and applying Boundary Value Analysis helps ensure robust input validation.

For enterprise-level applications requiring detailed auditing and secure transactions, SOAP APIs are more appropriate. SOAP provides a standardized protocol with built-in security features, ensuring data integrity and confidentiality. It also supports advanced transaction management and formalized contracts.

Continuous Integration (CI) involves the continuous merging of code changes, and this facilitates early detection of bugs through automated API testing. This ensures that potential issues are identified and resolved early in the development process.

REST API caching tends to be more complex due to its statelessness and varied caching mechanisms. Developers need to manage cache headers and expiration times, making implementation more intricate. SOAP, on the other hand, follows a standardized caching approach, simplifying the implementation process.

Including detailed error messages in API responses is crucial for effective error handling. Cryptic messages, random codes, or abbreviations can make it challenging for users or developers to understand and resolve issues. Thus, option b) "Detailed messages" is the appropriate choice for proper error handling.