You and your friend came close. Most of your problems, including the uninit issues, were the result of 'off-by-one' errors in array assignment, etc.

That said, this is also close (only "close" because the output formatting is NOT per your spec; correction is left as an exercise...) and utilizes what I believe to be some relatively minor corrections to your code.

#!/usr/bin/perl
use strict;
use warnings;

# 847021

=head OUTPUT OF THIS VERS:

>Long_DNA_Scaffold.0
atgctagsctgtagctagctagtcgatgctagtagct
ttagctagctgatgctagtcgatgctagtcatctagc

>Long_DNA_Scaffold.1

tagctagctagctgatcgtagctagtcgtagctagct
ggtcgatgctgatcgtagctgatgctagtcgatgcta
>                                       # removal of ">" and blank lin
+es left as an exercise....
>Long_DNA_Scaffold2.0
atgctagsctgtagctagctagtcgatgctagtagct
tagctagctgatcgtagctagctgatgctagctagtc

>Long_DNA_Scaffold2.1

tagctagctagctgatgctagctagtcgatcgtagct
ggctagtgctagtgctagctgatgctagtcgtagctg

=cut

my $file = "847021.txt";
open (IN, '<', $file) or die "Can't open $file: $!";    # test your fi
+le opens!

$/=">";
my $i=0;
my $separator_count=0;
my (@seq, $seq, @name, $name);

while (<IN>) {
    if(/^(\S+)?.*?\n(.*)/ms) {
        # $i++;                                         # this creates
+ an empty ("undef") element 0 in the arrays; mis-located; moved
        $seq[$i]=$2; 
        $name[$i]=$1; 
    } else {
        $separator_count++;
        if ($separator_count > 1) {
            print "No match for current data: $_\n\n";      # one inst
+ance is expected because the data begins with record separator; don't
+ show it
        } else {
            next;                                        # create no u
+ndef $arr[0]
        }
      }
$i++;                                                    # re-located 
+from line 57
}
close IN;
print "\n";

for my $j(0..$#seq) {                                    #   $#seq: in
+dex of last element. 0..$i replacing OP's 1..$i

    $name[$j] =~ tr/>//;                                 # replaces OP
+'s mis-coded tr//; see toolic's reply, above re tr syntax
    $seq[$j] =~ tr/>//;
    my @sequence=split(/N+/,$seq[$j]);

    my $n=length(@sequence);
    my$seqs_to_print=$n;                                 # OP's decrem
+ent caused failure to print second elements of @seq
    for my $k(0..$seqs_to_print){                        # use $i, $j,
+ $k for counters; $x and company are typically for substantive data  
    print ">$name[$j].$k\n";
    print "$sequence[$k]\n";
    }
    $j++;
}
[download]

Those comments above re good practice should be read as ADDITIONS to toolic's list.