Does Google just devalue spammy links, or does it devalue the links and penalize your site on top of it?
It will Devalue the links but also penalize your site if you have been a big abuser of unnatural links
Does it hit specific keywords, or does it have effects site wide?
It will do either of those depending on the severity and you will be able to see this in webmaster tools sometimes.
Does penguin run constantly, or only on specific days?
Penguin runs roughly twice a year currently, but is always in effect, you can get an unnatural links penalty at any time. Some have said it runs in some way with Panda
If spammy links are removed, how long does it take to recover?
If you have a Manual penalty you will need to file for a reconsideration request. Once your penalty has been revoked you will need to wait for the next Penguin refresh (My estimate is April/May)
If its an algorithmic issue, you can see changes happen once a month but you will need to wait for Google bot to crawl those changes. For example if you remove a link you will still need to wait for Google to crawl that site and see those links have been removed. And in some cases wait for a Algo update.