I noticed on the first day my iOS app hit the app store I got a download from Albania and another one from China. The app is only useful to a specific group, all of which are in a certain U.S. city. Are there bots that automatically download every new app? If so, to what end?
Yes, there are bots that auto down applications in order to boost rankings of applications:
Your downloads were most likely used as a way to mask the fact they are downloading specific apps to boost rankings. So 20 bots download Angry Birds (for example), 2 of those also download yours and some others to make it look like normal usage.
You could also account for these as simply random downloads, at a previous employer we created a very specific application to monitor solar panels, whilst they were only sold to specific people we had downloads (and subsequent uninstalls) from random countries around the globe.