When I used your code I added an open line called "Input" and exchanged my $line = <$fh> with my $line = <Input>. Ought I have done that?

The output, when I made that change, however looked exactly like the input file. If I gave a sample of the actual text would that help?

Anyway, I added some headers to your sample and put it into file 'input.txt':

section start marker
dear (13)
dear friends (22)
love (10)
dear friend (10)
loved (3)
dearly loved (1)
friends (1)

section start marker
competes in the games (1)
contend (1)
fight (2)
fought (1)
make every effort (1)
strive (1)
wrestling (1)
[download]

use strict;
use warnings;

open my $file, '<', 'input.txt' or die $!;

process_file($file);

exit 0;

sub process_file {
    my ($fh) = @_;

    while ( my $line = <$fh> ) {
        print $line;

        if ( $line =~ /section start marker/ ) {
            $line = handle_section($fh);
            redo if defined $line;
        }
    }
}

sub handle_section {
    my ($fh) = @_;
    my ( @entries, $line );

    while ( $line = <$fh> ) {
        last unless $line =~ m{
            ( [^(]+ ) # 1 anything except opening paren
            \s        #   space
            \(        #   opening paren
            ( \d+   ) # 2 number
        }x;

        push @entries, [ pack( 'Na*', $2, $1 ), $line ];
    }

    print map  { $_->[1] }
          sort { $a->[0] cmp $b->[0] } @entries;

    return $line;
}
[download]

section start marker
dearly loved (1)
friends (1)
loved (3)
dear friend (10)
love (10)
dear (13)
dear friends (22)

section start marker
competes in the games (1)
contend (1)
fought (1)
make every effort (1)
strive (1)
wrestling (1)
fight (2)
[download]

Ok, so I took this program and and that sample input.txt, ran it, and it worked. Now I do need to levels of sort, first descending numerical and then within the equal numbers, ascending alphabetically.

I've started reading on how to use map and am a little confused, but feel like I'm close.

In this section I know that I have to declare the first character in the string such that I can then sort by it, yes? This doesn't work, but it's something close?

sub handle_section {
    my ($fh) = @_;
    my ( @entries, $line );

    while ( $line = <$fh> ) {
        last unless $line =~ m{
            ( [A-Za-z] ) # 1 first character
            ( [^(]+ )    # 2 anything except opening paren
            \s           #   space
            \(           #   opening paren
            ( \d+   )    # 3 number
        }x;

        push @entries, [ pack( 'Na*', $2, $3, $1 ), $line ];
    }

    print map  { $_->[2] }
          sort { $b->[0] cmp $a->[0] 
                    ||
                 $a->[1] cmp $b->[1]
                } @entries;

    return $line;
}
[download]

Doesn't it already sort alphabetically? :)

You'll probably want to read about Schwartzian transform. Anyway, perhaps it will be clearer without pack

        last unless $line =~ m{
            ( [^(]+ )    # 1 anything except opening paren
            \s           #   space
            \(           #   opening paren
            ( \d+   )    # 2 number
        }x;

        push @entries, [ $2, $1, $line ];
        ...
        print map  { $_->[2] }
              sort {
                $a->[0] <=> $b->[0] # sort by $2
                ||                  # or if equal
                $a->[1] cmp $b->[1] # sort by $1
              }
              @entries;
[download]

Our @entries is an array of arrays. Each entry (inner array) has:
• index 0 - stuff matched by $2 (digits)
• index 1 - stuff matched by $1 (beginning of $line)
• index 2 - the $line itself
For example, one entry looks like this: ['1', 'wrestling', 'wrestling (1)']

So, we first sort by '1', then by 'wrestling'. Or we can pack '1' and 'wrestling' in just one string and sort by that... that's just a neat trick :)

map simply extracts the line from entry.