What are the pros and cons of using CLIENT_RESULT_CACHE_SIZE and RESULT_CACHE_MODE
What are the benefits? - Faster data access with less disk access.

How will the client side cache be kept in sync?  From the Oracle Call Interface Programmer's Guide:


  The client cache transparently keeps the result set consistent with
  any session state or database changes that can affect its cached
  result sets.
  
  When a transaction modifies the data or metadata of any of the
  database objects used to construct that cached result, invalidation is
  sent to the OCI client on its subsequent round-trip to the server. If
  the OCI application does no database calls for a period of time, then
  the client cache lag setting forces the next OCIStmtExecute() call to
  make a database call to check for such invalidations.
  
  The cached result sets relevant to database invalidations are
  immediately invalidated, and no subsequent OCIStmtExecute() calls can
  match such result sets. The OCI statement handles currently fetching
  from these cached result sets, at the time such invalidations are
  received, can continue fetching from this (invalidated) cached result
  set.
  
  The next OCIStmtExecute() call by the process may cache the new result
  set if there is space available in the cache. The OCI client result
  cache periodically reclaims unused memory.
  
  If a session has a transaction open, OCI ensures that its queries that
  reference database objects changed in this transaction go to the
  server instead of the client cache.
  
  This consistency mechanism ensures that the OCI cache is always close
  to committed database changes. If the OCI application has relatively
  frequent calls involving database round-trips due to queries that
  cannot be cached, (such as DMLs, OCILob calls, and so on) then these
  calls transparently keep the client cache up-to-date with database
  changes.
  
  Note that sometimes when a table is modified, a trigger can cause
  another table to be modified. OCI client result cache is sensitive to
  all such changes.
  
  When the session state is altered, for example, if NLS session
  parameters are modified, this can cause the query results to be
  different. The OCI result cache is sensitive to such changes and on
  subsequent query executions, returns the correct query result set. The
  current cached result sets are kept (and not invalidated) for any
  other session in the process to match; otherwise, these result sets
  get "Ruled" after a while. There are new result sets cached
  corresponding to the new session state.
  
  If the application must keep track of all such database and session
  changes it can be cumbersome and prone to errors. Hence, OCI result
  cache transparently keeps the result sets consistent with any database
  or session changes.
  
  The OCI client result cache does not require thread support in the
  client.


*Do we have an auto mode for RESULT_CACHE_MODE?* - The Oracle Database Reference shows that the two options for this setting are MANUAL and FORCE.

What is the recommended mode for this? - This would depend on what your needs are.

To understand the benefits of the result cache, look at the Validation of the Client Result Cache of the Oracle Call Interface Programmer's Guide referenced earlier.