You are very close to having what you want. The following slightly modified version of your program may help.

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

{
    my $gi = "";

    while (<DATA>) {
        chomp($_);
        $_ =~ s/\(|\)//xmg;
        if ( $_ =~ /LOCUS\s+(\S+)/ ) { $gi = $1; }
        elsif ( $_ =~ /\A\s+(\w+)\s+(\S+)\.\.(\S+)/ ) {
            print "\n\t\t$gi\t$1\t$2\t$3";
        } elsif ( $_ =~ /gene\=\"(\S+)\"/ ) {
            print "\tgene=$1";
        } elsif ( $_ =~ /\/transcript_id\=\"(\S+)\"/ ) {
            print "\tTranscriptID=$1";
        }
    }
    print "\n";
}

__DATA__
     gene        348634..348822
                     /gene="LOC100129305"
                     /note="Derived by automated computational analysi
+s using
                     gene prediction method: GNOMON. Supporting eviden
+ce
                     includes similarity to: 1 Protein"
                     /db_xref="GeneID:100129305"
     mRNA        348634..348822
                     /gene="LOC100129305"
                     /product="similar to C21orf94 protein"
                     /note="Derived by automated computational analysi
+s using
                     gene prediction method: GNOMON. Supporting eviden
+ce
                     includes similarity to: 1 Protein"
                     /transcript_id="XM_001714760.1"
                     /db_xref="GI:169215265"
                     /db_xref="GeneID:100129305"
[download]

This produces

                        gene    348634  348822  gene=LOC100129305
                        mRNA    348634  348822  gene=LOC100129305     
+  TranscriptID=XM_001714760.1
[download]

update: removed the setting of $\ as it is not necessary.

update2: I used perltidy to reformat your program. You might like to give it a try.