How do I know if a certain database has a standby db created?Check init parameter, LOG_ARCHIVE_CONFIG.
How do I break-up their links/connection if there is, so that the PROD will go free and will standalone?
They have PROD and Standby database as mentioned verbally. Ii was mentioned also that standby was not working anymore or has failed or has been put to active accidentally.When standby is behind and inconsistency in recovery, then you cannot put database in open(read-only) + Recovery.
In this regard, is the PROD still have dependencies on the "failed" standby Db? I mean is it holding back some files for the benefit of the standby Db?In maximum performance, there will be no dependency on standby database, of course it tries to do send redo/archives from primary to the all standby databases, In case of Maximum protection/availability then primary database is depended on the standby database, Because transaction has to commit at least one standby database then it will be applicable on primary database, BTW there is some differences in maximum protection and availability, I mentioned in brief.
How do I know if a certain database has a standby db created? How do I break-up the tie or their links/connection if there is, so that the PROD will go free and will standalone?You have to find out, when the last sequence applied on standby. you can check by below query