If I wish to install applications such as Skype, DropBox, etc however without registering for a Google account, is that possible? If so, how do I install them without a Google account?
Use ADB to install the applications.
Install the Android sdk ( https://developer.android.com/sdk/index.html ).
Enable USB debugging in Settings>Applications>Development. This process can be done even without enabling side-loading of apps. Open a command prompt window and type:
adb install app_name.apk
You could use other App stores like Amazon App Store or GetJar, but you would also need accounts on those.
You can side-load apps via downloading APK files on PC and then installing them from your SD card. Just try opening the APK files with a file manager like ES File Explorer or ASTRO File Manager which would give a option to open it with Package manager or Install.
But I assume from your question that you don't want to enable Installation of Non-market apps, then you don't get much choice. Even Amazon App store requires that feature enabled.
one can download APK files available on Google Play Store, without a Google account, by using APK Pure. it also allows downloading previous versions which may be helpful for people trying older devices, which need older app version for compatibility reasons.
additionally, one may choose to use a region-specific build.