MySQL table locks solution -> InnoDb / Partitions
key_buffer_size ?  RAM size?

On huge tables, indexes, especially if they are not "appended to", cause disk hits -- whether MyISAM or InnoDB.

var0, being an AUTO_INCREMENT, will be "appended to" the end.  Essentially no disk hits for it.  I can't tell about the other 4 indexes.  Let's say they are all very random.  This implies that each row inserted will need 4 disk hits.  On normal drives that means about 25 rows inserted per second.

Solutions...

RAID striping (0,5,6,10).  That will give you a factor of improvement.

SSDs.  $$$

PARTITIONing.  This may help.  It may also help with some SELECTs.  To discuss further, please reveal some of the semantics of the indexed fields.  (dates?  md5s?  names?)  Also provide some of the SELECTs, so we can optimize them at the same time.  The "partition key" must be part of any UNIQUE key.  This is not a problem with the AUTO_INCREMENT, but it may be with the other UNIQUE key.

Nothing (yet) says whether the Engine will make any difference.