You don't seem to have problem 1 in your code. As for problem 2, if a name matches in several places, all occurrences will be printed. If you want uniqueness use an hash. The following example illustrates the differences:

use strict;
use warnings;

my @names = ('joe', 'mary', 'francis');
my @lines = <DATA>;

my %lines_matched;
my @all_found;

foreach my $name (@names) {
    my $wanted = $name;

    foreach my $line (@lines) {
        if ($line =~ m/\b$wanted\b/i) {
            push @all_found, $line;
            $lines_matched{$line} = 1;
        }
    }
}

print foreach @all_found;
print "\nNow the uniques:\n";
print foreach keys %lines_matched;

__DATA__
Joe is lazy
Robin Wood
Mary Poppins
Francis Bacon
Ruppert
Joe married Mary
[download]

outputs

Joe is lazy
Joe married Mary
Mary Poppins
Joe married Mary
Francis Bacon

Now the uniques:
Francis Bacon
Joe married Mary
Joe is lazy
Mary Poppins
[download]

update. added \b after reading kyle's comment.