If your data is as regular as it appears to be, a couple of splits instead of regexes will get the job done.

#!/usr/bin/perl -w
use strict;

while (<DATA>)
{
   chomp;
   my ($basedir, @dot_names) = split(/\./,$_);
   my $type = (split('/',$basedir))[-1];

   my $school = "null";  
   $school=$dot_names[-2] if (@dot_names >1);
  
   #open the file here and use the variables as prefix for each line
   print "type: $type  school: $school\n";  
}
=prints 
type: abc  school: null
type: def  school: null
type: abc  school: cole
type: abc  school: drew
type: def  school: cole
type: def  school: drew
=cut

__DATA__
/home/test/abc/.date_run_dir
/home/test/def/.date_run_dir
/home/test/abc/.date_file_sent.email@wolverine.cole.edu
/home/test/abc/.date_file_sent.dp3.drew.net
/home/test/def/.date_file_sent.email@wolverine.cole.edu
/home/test/def/.date_file_sent.dp3.drew.net
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