Does Salesforce offer their Salesforce1 APK for developers to download? It seems like the only places I can find it are third-party websites and I would be much more comfortable downloading it from Salesforce itself.
Officially, most vendors do NOT provide APKs directly. But unofficially, there are multiple ways to get it :)
A) You can actually get a Android Simulator VM (plenty of such options on the Internet) which runs Google Play and then download the app that you need
B) You could use the plugin quoted on this thread to pull the APK down from Google Play
Having said that, it's still recommended to be using the official APK on a device.