icmp ping requires root privilege

I don't think there's a way to 'fix' this...  The usual command line utility ping has the same problem, which is why it's set-user-id-root:

$ ls -l `which ping`
-rwsr-xr-x 1 root root 37656 Apr 23  2006 /bin/ping
   ^         ^
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$p = Net::Ping->new("tcp");
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It would have been better to enclose your code snippet in "code" tags. Please read Writeup Formatting Tips. It's not too late to edit your post, deleting all "br" tags, and adding "code" tags around your code.

- what is the "2" in print "$host is dead\n" unless $p->ping($host, 2);?

$p->ping($host [, $timeout]);
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Ping the remote host and wait for a response. $host can be either the hostname or the IP number of the remote host. The optional timeout must be greater than 0 seconds and defaults to whatever was specified when the ping object was created.

is there any shortcut way to put this in the @array?

my @array = qw(4.4.4.4 127.0.0.1 123.333.333.333);
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Another general tip: add these to the top of your code:

use warnings;
use strict;
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I have no answer for your 1st question :(

foreach $host (@host_array) {
    if( $p->ping($host, 2) ) {
        print "$host is alive.\n";
    }
    else {
        print "$host is dead\n";
    }
}
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Speaking of ranges of IP addresses and needing root access for ICMP ping... Ping scanning your own network can be a useful tool, but its usefulness is limited ¹. Scanning someone else's network is more than rude. It may constitute a breach of acceptable use policies or terms of service contracts. It may even be illegal under certain circumstances in certain jurisdictions, but I'm not a lawyer so I couldn't say where or under what circumstances it would be. Ping flooding a machine or a range of addresses is even worse.

Pinging requires root access for the same reason many other things do. It is potentially destructive if used incorrectly. Under normal circumstances and for proper use, it's handy and benign. Please use it properly.

¹ If you need to see what addresses are assigned by your DHCP server, there should be a way to query that server rather than pinging your whole network. If you have static assignments, they should go in a database as they are assigned. Any attacker putting a machine on your network that they want hidden will prevent that machine from responding to pings, so you'd only catch a real novice script kiddie.

If I want to have IP addresses like: 1.1.1.0/8 or 203.10.22.19 to 203.10.22.255 is there any shortcut way to put this in the @array?

sub get_ips
{
    my $subnet_specification=shift;                            # Get t
+he subnet specification.
    my ($network, $mask)=split(/\//, $subnet_specification);          
+  # Break it into a subnet and mask.
    my @subnet_array=split/\./,$network;                        # Spli
+t the network into individual octets.
    my $subnet_value=$subnet_array[0]*0x1000000+$subnet_array[1]*0x100
+00+$subnet_array[2]*0x100+$subnet_array[3];
    # $subnet_value now contains a 32-bit value corresponding to the n
+etwork address given.
    # Now calculate the top end of the subnet.
    my $top_address=$subnet_value+(2**(32-$mask)-1);
    # We now have the first and last addresses required. Produce a lis
+t of all addresses in between, in dotted-decimal format.
    my @return_value;                                # Holds the array
+ of addresses we are returning
    
    for (my $address=$subnet_value; $address<=$top_address; $address++
+)
    {
        my @octets=(int($address/0x1000000)%0x100, int($address/0x1000
+0)%0x100, int($address/0x100)%0x100, $address%0x100);
        push(@return_value, join(".", @octets));                    # 
+Assemble it into a dotted-decimal form and add it to the returned arr
+ay.
    }
    return @return_value;
}
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