$dbm{'name'} = $name overwrites the previes entry
# Yes, quite correct

$dbm{$name'} = $name creates redundant Aaron Aaron
# I don't think so. You've got bad punctuation there,
what do you expect Perl to do? Perhaps you're creating a
 value of "Aaron" but with a key of "Aaron'" -- see quotes?

$dbm{$name}; creates Useless use of DBM error
# quite so. Meaningless if that's the whole statement.

$dbm{$name} = ""; writes Aaron
# no it doesn't, I'm betting. $dbm{$name} should be an
 empty string now.