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.

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.

The selection of Data Nodes is essential in determining the processing power of a Hadoop cluster. Data Nodes are responsible for storing and processing data, and the number and capacity of these nodes significantly impact the overall processing capabilities of the cluster.

ZooKeeper plays the role of configuration management in the Hadoop ecosystem. It is a distributed coordination service that helps manage and synchronize configuration information across the cluster, ensuring consistency and reliability in a distributed environment.

MapReduce optimizes the processing of large datasets through data partitioning. This mechanism involves dividing the input data into smaller partitions, with each partition processed independently by different nodes. It facilitates parallel processing and efficient resource utilization in the Hadoop cluster.

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.