I know this is a elementary question, but I figured I'd get a quicker answer here than Apple support. Where can I find out how many times the app has been downloaded and other download stats? I'm a developer and can't find the stats on the iOS Dev Center. I have not renewed the $99 Developer license, so could it be hidden because of that?
If it is the application you developed, you can access it from itunesconnect.apple.com which is called iTunes Connect, and that's how you should have uploaded your app to Apple in the first place.
You can also try a piece of software called AppFigures, which is essentially a portal to iTunes Connect, which I find easier to use and with more features.
Also: after not renewing your iPhone Developer licence, your apps will stay in iTunes Connect for some period of time, probably around a year before being deleted.
"The App". Your app? Any app?
You cannot find out how many times any arbitrary app has been downloaded.
Your own apps, yes. With an in-good-standing (paid up) developer account, you will be able to access iTunes Connect, which offers some download metrics, going back thirteen weeks into the past.