To handle multiple fields, I have the hash store a hash of values, rather than a single value. That way I can access the fields by name. The top level is still controlled by coordinate name. The changes from the previous version are minor:

use warnings;
use strict;

open my $data, '<', 'alldata.txt'
    or die "Could not open 'alldata.txt': $!\n";

my %dataset;

RECORD:
while ( my $line = <$data> ) {
    chomp $line;
    next RECORD
      unless $line =~ m{\d};
    my ( $coord, $dist, $chr, $exons, $pals ) = split /\s+/, $line;
    next RECORD
       if      exists $dataset{$coord} 
          and $dataset{$coord}{dist} < $dist;

    $dataset{$coord} = { dist        => $dist,
                         chr         => $chr,
                         exons       => $exons,
                         palindromes => $pals,
                       };
}

say "coord\tdist\tchr\texons\tpalindromes";


say "$_\t$dataset{$_}{dist}\t$dataset{$_}{chr}\t$dataset{$_}{exons}\t"
  . "$dataset{$_}{palindromes} "
    for sort { $a <=>  $b } keys %dataset;
[download]

• I just cut-and-paste your dataset, rather than converting to comma-separated, so I split on the regexp /\s+/ ... one or more spaces.
• I had blank lines at the end of the file. Rather than modify the file, I ignore lines that don't contain numbers. It's frequently useful to also skip lines that contain a '#', that way you can have headers and comments in the data file.
• I create an anonymous hash of all the data other than the coordinate, and then use the coordinate to identify the top level entries. Note also that you can add more comparison statements, for example if you have two records tied for distance, you could test for odd number of exons and do a next RECORD on that
• I didn't bother with a format string, too much counting. But I did split the string across two lines, and use a '.' to join them ... I don't like over-long code lines.

The entire data structure, according to Data::Dumper:

$VAR1 = {
          '6700' => {
                      'palindromes' => '4',
                      'chr' => '13',
                      'exons' => '1',
                      'dist' => 50
                    },
          '4678' => {
                      'palindromes' => '0',
                      'chr' => '6',
                      'exons' => '2',
                      'dist' => '45'
                    },
          '2346' => {
                      'palindromes' => '567',
                      'chr' => '12',
                      'exons' => '1',
                      'dist' => '78'
                    },
          '5349' => {
                      'palindromes' => '14',
                      'chr' => '8',
                      'exons' => '2',
                      'dist' => '6'
                    },
          '3456' => {
                      'palindromes' => '5',
                      'chr' => '10',
                      'exons' => '1',
                      'dist' => 67
                    },
          '567' => {
                     'palindromes' => '8',
                     'chr' => '5',
                     'exons' => '7',
                     'dist' => '344'
                   },
          '1345' => {
                      'palindromes' => '123',
                      'chr' => '5',
                      'exons' => '8',
                      'dist' => '567'
                    },
          '8964' => {
                      'palindromes' => '8',
                      'chr' => '2',
                      'exons' => '18',
                      'dist' => '560'
                    }
        };
[download]

P.S. The debugger is your friend: perl -d program.pl.

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