Your script (slightly modified) works for me:

use strict;

use Archive::Zip qw(:CONSTANTS :ERROR_CODES);
use File::Temp;
warn "Using Archive::Zip $Archive::Zip::VERSION";

my $tmp = $ENV{'TEMP'}.'/zip/extractMember';

my $zip = Archive::Zip->new();
if ($zip->read('tmp2.zip') != AZ_OK)
{
  die 'Einlesen klappt nicht.';
}

foreach my $member ($zip->members())
{
  print($member->fileName()."\n");
  my $target = File::Temp::tempnam($tmp, 'XXXXXXXXXXXXXXXXXXXXXXXXXXXX
+XXXX');
  print "First";
  $member->extractToFileNamed($target);
  print "\n";
  print "Second";
  $target = File::Temp::tempnam($tmp, 'XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
+X');
  $member->extractToFileNamed($target);
  print "\n";
}
__END__
Using Archive::Zip 1.23 at tmp.pl line 5.
tmp2.txt
First
Second
[download]

Note that I'm using an older version of Archive::Zip here. I also don't get the warning about the CRC32 mismatch:

format error: CRC or size mismatch while skipping data descriptor
at C:/Programme/Perl/lib/Archive/Zip/ZipFileMember.pm line 189
[download]

Are you sure that your archive is OK? I created my test archive using 7zip.

I'll be upgrading to Archive::Zip 1.30 and retest.

Update: Upgraded to Archive::Zip 1.30 and it still works for me:

Using Archive::Zip 1.30 at tmp.pl line 5.
tmp2.txt
First
Second
[download]

My guess is that your zip file is broken.