Problem with MySQL MyISAM — duplicate entries and crash on large tables

Too many connections — part I will explain. On myisam you can't do it as you described. Active block selects the entire table and the inserts are waiting for the lock. So, if your select stopped for a second, then your data in the queue has already gotten 1000 inserts. Here you have too many connections.
Switch to InnoDB, it is slow, but there is blocking at the row level and failure, she recovered well.
Better to make one extended insert 1000 rows than 1000 inserts one row. Think about it.

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Notice for those who fiercely shit bricks higher, at 70 million records in the largest table is not so much. Not much to say, "for Example, if in one table a lot of records (70 million), it is possible to use the partition".
\r
7 billion and tens of terabytes total size of the database — that is a lot.

with a large number of inserts of your choice innodb

It is necessary to look locks in mysql and iostat at all, and that there may be disk related.
And generally somehow it is strange to see MyISAM for tables that are frequently written.