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.
Many android devices which do not support the Google Play store DO support Amazon Appstore, which functions much the same. You can browse or search thousands of android apps on Amazon's site, or via the app itself, and upon finding one you like, you can install it directly, just as you would via Google Play. You can purchase paid apps using your amazon account, as well as download many free ones. There are other apps of this nature as well, such as GetJar.
I made a stupid mistake on the other device is called Plays Store and here it is called simply Market, anyway I found you can also download it as an .apk file and install it.
