My assumption is if the shebang is not mandatory for pm files in Linux (because pm files are not meant to be executed from command line), then I guess the file's permissions can be 644. They are not required to be 755. Correct?

Yes!

My assumption is if the shebang is not mandatory for pm files in Linux (because pm files are not meant to be executed from command line), then I guess the file's permissions can be 644. They are not required to be 755.

Yes. For cheap thrills, I ran these commands on my recently installed perl 5.38 on Ubuntu:

~/localperl$ find . -name '*.pm' -ls >pm.tmp
~/localperl$ find . -name '*.pl' -ls >pl.tmp
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All the .pm files have permissions like this:

-r--r--r--   ... ./lib/5.38.0/Carp.pm
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The perl commands in the bin directory designed to be run as commands have permissions like this example of the prove command:

-rwxr-xr-x  ... prove
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These text files also tend to start with a shebang line like:

#!/home/youruser/localperl/bin/perl
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There are also plenty of .pl files lying around with permissions like this:

-r--r--r-- ... somefile.pl
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perl /somepath/somefile.pl
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There is one trick that I sometimes use....
It is possible for your module to know if it was "used" versus executed directly. This allows me to put a test driver in the module itself that I use during development that presents more info that just a "pass/fail". If you do this, you will want your .pm to be an executable file under Unix.

#Normal module preamble here...
sub test{
   print "this is a dev test...\n";
}

test() if not caller; ### Run test() if run directly

#Normal rest of module code here....
1;
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Oh, what a nice trick! That's so cool!! :-)