If I understand the problem correctly, then you might try approaching it like this:

#!/usr/bin/perl -w
use strict;

our ($File1, $File2) = qw/file1 file2/;
open File1 or die "$File1: $!\n";
open File2 or die "$File2: $!\n";

my ($key, %results);
while (<File1>){
    next if /^\s*$/;
    chomp;
    if (/^>\s*(\S+)/){
        $key = $1;
    }
    else {
        $results{$key} = [ split ];
    }
}
close File1;

my ($len, $side, $str);
while (<File2>){
    next if /^\s*$/;
    if (/^>([^_]+)_(left|right).*?(\d+)\s*$/){
        print;
        $str = $1;
        $side = $2;
        $len = $3;
    }
    else {
        my @list;
        @list = @{$results{$str}};
        if ($side eq 'left'){
            die "$str is too short for a left slice of $len!\n"
                unless @list >= $len;
            print "@list[0..$len-1]\n";
        }
        else {
            die "$str is too short for a right slice of $len!\n"
                unless @list >= $len;
            print "@list[@list-$len..$#list]\n";
        }
    }
}
close File2;
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