I didn't get the line:

die sprintf "ERRRR(%d)(%s)(%d)(%s)", $!,$!,$^E,$^E;

Nor did I get it to work, putting it in as the last line of my code.

ERRRR(9)(Bad file descriptor)(0)() at C:\SCRIPTING\URLTESTv6.pl line 22.

As for these redirected websites, the one in question http://www.eserve.com.sa is HTTP. In any case, should I conclude that redirection isn't supported. In that case I'll need to hardcode the eventual address.

At the end of your code its too late. You're supposed to get more info about which Unknown Error it is. Try

#!/usr/bin/perl --

use strict;
use warnings;

use LWP::UserAgent 5.829;

@ARGV = qw[  http://www.eserve.com.sa/ ];

my $url       = shift or die "URL expected\n";
my $useragent = LWP::UserAgent->new;

$useragent->proxy(['http', 'ftp','https'], 'http://192.168.5.39:8080/'
+);
$useragent -> timeout( 20 );

my $request   = HTTP::Request->new( GET => $url );
my $response  = $useragent->request($request);
my $err =  sprintf "(%d)(%s)(%d)(%s)\n", $!,$!,$^E,$^E;
print $useragent->proxy('http'),"\n";
print "Err : $err\n";
print $response->dump,"\n";

__END__
lwp-request -useUS -m head http://www.eserve.com.sa/
lwp-request -useUS -m get http://www.eserve.com.sa/
[download]

The number of possible unknown errors is huge (goes into 11_000), to see some run

for my $i( 0 .. 256 ){ # 
    $! = $i;
    $^E = $i;
    printf "(%d)(%s)(%d)(%s)\n", $!,$!,$^E,$^E;
}
[download]

As for these redirected websites, the one in question http://www.eserve.com.sa is HTTP.

Which always redirects to the HTTPs version.

In any case, should I conclude that redirection isn't supported.

No.