Those messages are warnings, so if you really don't want to see them you can bracket your use statements like this:
no warnings;
use WWW::Mechanize;
# ... other use statements here
use warnings;
But I don't advise that because you'll miss out on messages like "Constant subroutine StevesPerlTools::DATA_DIR redefined" which sounds to me like something requiring fixing. If the third-party modules on their own compile without warnings then again these likely indicate some problem with the way you are using them. OTOH, if the warnings persist when the modules are compiled stand-alone that's probably worth bringing to the attention of the module author.
If there are warnings or errors reported from the syntax check just hit return and it will take you to where the error occurred in your code. Fix that, re-run the syntax check and the other window will refresh with the new report. Keep doing that until they're all gone and you'll simply have the line "file.pl : Syntax is OK" in your status line.