Here's a bit-logic based approach:

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

my %groups = ( 
    group1 => ['cooling', 'rooting', 'hooting', 'looking', 'doormat', 
+'cooking', 'cookies', 'noodles'], # desired result "-oo----"
    group2 => ['yourself', 'southern'], # desired result "-ou--e--"
    group3 => ['arden', 'arlen', 'asked'] # desired result "a--e-"
);

sub show_bits {
    #print unpack("B*", shift), "\n";
}

for my $g (values %groups) {

    my $w1 = $g->[0];
    my $and = "\xff" x length($w1);
    my $or  = "\0"   x length($w1);
    
    for my $w (@$g) {
        print "$w\n";
        #print join(" " x 7, split //, $w), "\n";
        show_bits($w);
        
        $and &= $w;         show_bits($and);
        $or  |= $w;         show_bits($or);
    }
    
    my $xor = $and ^ $or;   show_bits($xor);
    $xor =~ tr/\0/\xff/c;   show_bits($xor);
    my $mask = ~$xor;       show_bits($mask);
    
    my $common = $w1 & $mask;
    $common =~ tr/\0/-/;
    print "$common\n";

    my $i = -1;
    print map {$i++; $_ ne "-" ? "$_ : $i\n" : ()} split //, $common;
    print "\n";
}

__END__
yourself
southern
-ou--e--
o : 1
u : 2
e : 5

cooling
rooting
hooting
looking
doormat
cooking
cookies
noodles
-oo----
o : 1
o : 2

arden
arlen
asked
a--e-
a : 0
e : 3
[download]

Uncomment the commented out prints to see how the bit strings are being transformed/combined...

(Update: fixed cut-n-paste error $and ^= $or — thanks johngg)