Kerberos in Hadoop provides authentication, ensuring that only authorized users can access the Hadoop cluster. It uses tickets to verify the identity of users and prevent unauthorized access, thus enhancing the security of the Hadoop environment.

The number of Mappers in a custom MapReduce job is primarily determined by the size of the input data. Each input split is processed by a separate Mapper, and the total number of Mappers is influenced by the size of the input data and the configured input split size.

In Hadoop administration, HDFS Balancer is essential for balancing data and processing load across the cluster. The HDFS Balancer utility redistributes data blocks across DataNodes to ensure uniform data distribution and prevent data imbalance.

In Impala, Data Caching is a mechanism that speeds up data retrieval operations. Caching involves storing frequently accessed data in memory, reducing the need to read from disk and improving query performance. It is particularly useful for repetitive queries on large datasets.

Hive utilizes SerDes (Serializer/Deserializer) to process data that is not structured in a traditional database format. SerDes allow Hive to interpret and convert data between its internal representation and the external format, making it versatile for handling various data structures.

Logging is a common practice in debugging Hadoop applications. Developers use logging statements strategically to capture information about the flow and state of the application at various points. This helps in diagnosing issues, monitoring the application's behavior, and improving overall performance.

For advanced data processing in Hadoop using Java, the Apache Spark API provides more flexibility than traditional MapReduce. Spark offers in-memory processing, iterative processing, and a variety of libraries, making it well-suited for complex data processing tasks.

The Java-based API commonly utilized to interface with Hadoop's HDFS is the HDFS API. This API allows developers to interact with HDFS programmatically, enabling tasks such as reading and writing data to the distributed file system.

Apache Pig is a high-level scripting language built for Hadoop that excels at complex data transformations and aggregations. It provides an abstraction over MapReduce and simplifies the development of intricate data processing tasks. Pig's ease of use and flexibility make it suitable for scenarios requiring complex data transformations.

ZooKeeper plays a crucial role in maintaining high availability by providing coordination services. It helps in synchronizing distributed processes and managing configuration information, making it easier to handle failover scenarios and ensuring that the Hadoop cluster operates smoothly.