Would it be considered Bad Form to toss all of these into a Common.pm file and then require them?

IMHO, no (not that it makes much difference since you can only ever require a module once (without munging %INC, of course)). However this won't work for lexically scoped pragma such as strict and warnings, as they will only apply their particular pragmatic effects to the given lexical scope. So you will have to explicitly include the likes of strict into your desired modules. You'd also want to make sure that the the importing from the modules in Common is bubbled up to the caller's module e.g so the Dumper function from Data::Dumper can be called without fully qualifying it (i.e it's been imported into the caller's package).

Would each package get their own copy of $log with the correct package name?

Again, because of lexical effects $log will not be visible in external modules. In this particular case you could either use a function such as Common::log to return a lexically scoped $log or just have a $log package variable e.g $Common::log. Search around for singleton objects for more information on the subject.
HTH

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broquaint