How do I tie Flurry performance data to Yahoo Native? In the documentation, it looks like this API is for Yahoo campaigns. Do I need a different advertiser ID?

You need to know the advertiser ID of your Yahoo Native account and pull data from the campaigns that belong in that account. Technically, these are Yahoo Native video app install campaigns, but they apply to all types of mobile campaigns that run on the Flurry network.

Is Flurry campaign data mixed in with Yahoo Native data, or can I cleanly separate these sources?

Data can be separated at the advertiser ID level, campaign level, and so on, as described in the Yahoo Native API documentation.

Why does the data show up in the UI at a considerable amount of time ahead of the API?

The Native UI has more up-to-date data than the API. However, the UI data is directional only and is likely to change during the close of books for the day. The data from the API is more final. You can use the current APIs for intra-day updates.