Do you have a perltidy install that works for you?

Yes, I do.

So, so far, I've got nothing.

When I run perltidy on your code, I get a new file named pt1.pl.tdy, as described by:

$ perldoc perltidy
[download]

When I run diff on the 2 files, I see many whitespace differences, as expected:

$ diff pt1.pl pt1.pl.tdy
7,9c7,9
< my $m = WWW::Mechanize->new( qw/ autodie 1 /); 
< $m->get( $domain);
< my @list = $m->images();
---
> my $m      = WWW::Mechanize->new(qw/ autodie 1 /);
> $m->get($domain);
> my @list    = $m->images();
11c11
< my $dir = &DirPP;
---
> my $dir     = &DirPP;
15,16c15,16
<     my $filename = "$dir". "/image_". "$counter";
<     getstore($url,$filename) or die "Can't download '$url': $@\n";
---
>     my $filename = "$dir" . "/image_" . "$counter";
>     getstore( $url, $filename ) or die "Can't download '$url': $@\n"
+;
19c19
<     sub DirPP {
---
> sub DirPP {
21,32c21,29
<         use Errno qw/ EACCES /; # permission denied
<         state $counter2 = 1;
<         my $word    = "site";
<         my $name ;
<         my $made = 0;
<         while(1){
<             $name = sprintf '%s%3d', $word, $counter2;
<             last if not $made = mkdir $name, 0755 ;
<             die $! if $! == EACCES;
<         }
<         return $name if $made;
<         return;
---
>     use Errno qw/ EACCES /;    # permission denied
>     state $counter2 = 1;
>     my $word = "site";
>     my $name;
>     my $made = 0;
>     while (1) {
>         $name = sprintf '%s%3d', $word, $counter2;
>         last if not $made = mkdir $name, 0755;
>         die $! if $! == EACCES;
33a31,33
>     return $name if $made;
>     return;
> }
35,37c35
< 
< 
< $ 
---
> $
[download]

q2) Can someone discuss the difference between Perl::Tidy and perltidy()? Are they related under the hood, for example?

Perl::Tidy is used by perltidy. perltidy is just a wrapper script:

$ perldoc -m perltidy
[download]

If you want to edit the file in place, leaving a backup file, use perltidy -b

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