MySQL binary log - write operation
Here are the answers to each of your question


  1 How the copying of binary log works?


I wrote a detailed list of the steps taken by MySQL Replication in my answer to  Is MySQL Replication Affected by a High-Latency Interconnect?.

Someone else answered that same post, but from the point of view of DB Client


  2 Is it first writes to master binary log and copied to destination log?


There are many middle steps


Master completes a SQL update (single update or group of updates (AKA group commit))
Master writes one of the following to its binary log


the SQL command if binlog_format=STATEMENT
the row changes if binlog_format=ROW

IO Thread from the Slave copies the SQL Update from Master's binary log to its relay logs
SQL Thread on slave reads updates from its relay log and processes. If log_slave_updates is enabled, then the Slave writes that SQL command to its local binary log.



  3 Is it writing on both the source and destination binary log simultaneously?


Never. Not with semisync replication, not with GTID replication either. Just look at the above steps


  4 What will be impact of MySQL performance and disk IO performance?


That depends where binary logs and relay logs are stored. If data and logs are on the same disk mount, you will both random writes for data and sequential writes for logs. Both will be competing for the same disk space. You should separate them (See my post from last year Should the Redo log on a production MySQL server be configured to another location?)