Rebuild index online without risk to other requests
There are several options available to help reduce the problems you're seeing:

/* set the LOCK_TIMEOUT value to the maximum number of milliseconds 
   you want to wait for the ALTER INDEX REBUILD to start
*/
SET LOCK_TIMEOUT 100;

/* this tells SQL Server our ALTER INDEX REBUILD statement should
   be the deadlock victim
*/
SET DEADLOCK_PRIORITY LOW;

/* Rebuild the index */
ALTER INDEX MyIndex 
ON MySchema.MyTable 
REBUILD WITH (
    ONLINE = ON
    , ALLOW_ROW_LOCKS = ON
    , ALLOW_PAGE_LOCKS = ON
    , MAXDOP = 1
    );


I know you have SQL Server 2012, however with SQL Server 2014+, you can use the WAIT_AT_LOW_PRIORITY option to tell SQL Server to allow other sessions to proceed while the online index build operation is waiting to obtain its schema modification locks.

ALTER INDEX MyIndex 
ON MySchema.MyTable 
REBUILD WITH (
    ONLINE = ON (WAIT_AT_LOW_PRIORITY (MAX_DURATION = 1 MINUTES, ABORT_AFTER_WAIT = SELF))
    , ALLOW_ROW_LOCKS = ON
    , ALLOW_PAGE_LOCKS = ON
    , MAXDOP = 1
    );


From the MSDN SQL Server Books Online:


  In order to execute the DDL statement for an online index rebuild, all active blocking transactions running on a particular table must be completed. When the online index rebuild executes, it blocks all new transactions that are ready to start execution on this table. Although the duration of the lock for online index rebuild is very short, waiting for all open transactions on a given table to complete and blocking the new transactions to start, might significantly affect the throughput, causing a workload slow down or timeout, and significantly limit access to the underlying table. The WAIT_AT_LOW_PRIORITY option allows DBA's to manage the S-lock and Sch-M locks required for online index rebuilds and allows them to select one of 3 options. In all 3 cases, if during the wait time ( (MAX_DURATION = n [minutes]) ) there are no blocking activities, the online index rebuild is executed immediately without waiting and the DDL statement is completed.