How to install Android apps without having Google Play? If you have Google Play application installed is very easy to find and install an application. But some devices seem not to have Google Play application installed. Maybe you can use Google Play in browser to search for those applications.

Is there a way you can find and download the .apk files if you know the application name without rooting and installing Google Play application?
What you're referring to here is called sideloading - this is a term used when loading an app on to a device without using the/a market store.

It is possible to get APK files for apps, but there're generally good reasons why most developers don't make these publicly available - some developers will make these available thought, it depends on the individual.  

I would email the developer in question and ask if you could have an APK version of the app.  I wouldn't imagine that they would have any issues with this if your request is legitimate and above board.