It seems, to me, given the above structure, that since the data i spread over multiple lines. the solution lies in using a flip-flop operator e.g.

use warnings;
use strict;

my $save;

while (<DATA>) {
    if (/^\{TAG\}/ .. /^\{COUNT\}/) {
        unless (/^\{(TAG|COUNT)\}/) {
            $save = $_;
            print "{TAG} and {COUNT2} is 1\n" if /\sand\s/;
        }
    }
}

__DATA__
{TAG}
0012310002 and 0012310003
{COUNT}
000000
{COUNT2}
000000 and 100001
[download]

Produces (not entirely unexpectedly:-):

$ perl tst.pl
0012310002 and 0012310003
$
[download]

Update:

Arrrgh!! Omitted 2nd question - now included. Also didn't show how to save the data between the given tags - also now included.