As for your second question,

• You close PDF before you read from it.
• You binmode when you read, but you don't when you write.
• print while <PDF>; would use less memory.
• What's $output?
• It's better to use the 3-arg open over the 2-arg open to open files.

my $file_name = "LeasevsBuy.pdf";
open(PDF, '<', $file_name)
   or die "could not open PDF \"$file_name\": $!\n";
binmode PDF;

my $size = -s $file_name;

print "Content-Type: application/pdf\n";
print "Content-Length: $size\n\n";
binmode(STDOUT);
$/ = \4096;  # Read 4k at at time.
print while <PDF>;

close(PDF);  # Optional.
[download]

Update: Overlooked that $output was used without being set. Fixed.