Ok, brain dump of a moderator (and very active user on Workplace, where we have similar concerns).
First, don't treat symptoms - the root cause is poor question quality and a community ok with poor quality answers. A clear question is a lot easier to answer meaningfully than a bad one. Take care to close poor questions first, this will help a ton.
Downvote poor quality answers and suggest improvements. Some people will be hopeless, sure. But most people writing answers like it and want to be helpful. But most of the Internet is a barren wasteland though, where you drop thoughts and don't have to bother with explaining why.
How many sites take moderation action when answers are posted without sources (the one I'm most familiar with is Skeptics). What form does this action take (is the post removed, is a post notice added, etc.)
My rule of thumb is, "if you don't explain why you would do this, it's not backed up." This can be a variety of forms. Oftentimes on Workplace, it's just explaining why. This meta post probably sums my perspective up the best I can think of.
Regarding actions, generally a post notice and a comment (the Workplace community does a great job with this, mods rarely have to do this) serve to be sufficient. Though in more dramatic cases answers can be deleted.
How many sites take unofficial action on answers without sources (i.e. downvotes, comments linking to a meta answer, etc.)?
When is it a good idea to transition from taking unofficial action to taking official action on answers without sources?
Depends. This is really a hard question to answer. Some people care about downvotes/comments. Some don't. Some don't care about post notices (you don't get a ping for one as far as I know).
Generally I wait until I see the user has been online a while after a comment/post notice and no action before doing anything officially.
How much addition work does a sources policy create for moderators?
Depends on how the site feels. Sites should self moderate. Moderators should not be the only ones enforcing this - if anything, moderators should only handle the edge cases. Review queues and downvoting/commenting are the easiest way to do this as a site.
There are always people who want to post whatever they want and not post any reasoning/backing. You have to accept this. As a site and moderator you are a teacher and representative, you will encounter this in some fashion and have to gently guide people into that expectation.
Right now, I think minimal work on Workplace. But that's because as a community the site downvotes answers without sources, etc. The initial stages were harder I think (I was a high rep user for much of the time getting the community to support this overall, perhaps an older or ex-moderator can discuss this more detailed for the process of getting there).
Does a "sources are required" policy significantly increase the quality of answers? In other words, is it worth the time to implement one?
The policy doesn't affect it at all. Site culture does ;)
I would suggest focusing on question quality though, first. Good questions will tend to get good answers. Bad questions normally get bad answers.
Requiring sources to be applied when most questions are crap won't be fruitful and will be... painful.
What are the downsides to a sources-are-required policy?
The entire Internet is a forum. Stack Exchange is not. You will face resistance from people who want a site to be a glorified discussion board. Accept this, and realize you can't please everyone.
The biggest downsides are popular opinions, without any explanation. Such as "just quit" - this might be a popular answer, but on a site like Workplace this has significant potential impact to people's lives (adding incorrect code can be undone trivially; unquitting a job cannot).
These are the hard cases. Especially when it's a popular answer that doesn't explain the "why?" or even answer the question..