Replicating from MySQL 5.1 master to 5.6 slave failing because 'INSERT ... VALUES (NOW())' results in 'Error_code: 1062'
Can I assume that 'INSERT INTO "Member" ("Created") VALUES (NOW())' is
  triggering the error here?


Yes that's exactly what the error telling

Error 'Duplicate entry '8072' for key 'PRIMARY'' on query. 
Default database: 'DATABASE'. Query: 'INSERT INTO "Member" ("Created") VALUES (NOW())', 
Error_code: 1062



  Can I get replication to work without skipping rows with SET GLOBAL
  sql_slave_skip_counter = 1;?


Yes, You can go and delete the row with the ID 8072 in the table.. you can add the option of slave-skip-errors = 1062 in my.cnf and restart mysql

But you have to be extremely wary about doing this, as the referential integrity is there for a reason, and without it, your slave could not be satisfactorily promoted to take the place of the master in the case of something going wrong with your master hardware...

The only clean way is:

In the master 


You FLUSH TABLES WITH READ LOCK; in the database you want to
replicate
SHOW MASTER STATUS and copy the file name and the position
Excute a full backup to the database you want to replicate
Then release the tables with UNLOCK TABLES;


In slave


Restore the databse
Excute this CHANGE MASTER TO MASTER_HOST = 'masterhost', MASTER_USER
= 'masteruser', MASTER_PASSWORD = 'masterpass', MASTER_LOG_FILE = 'filename', MASTER_LOG_POS = 'position';
Then start slave;