My Android Studio does not recognize any of the sample apps with Force.com Android SDK as a valid project. What steps do I need to take to open and run it there?
This is because Gradle support was actually introduced in Mobile SDK 3.0 which was released in January end. Please upgrade to 3.1 and you should be good to go.
The 3.0 release notes states
Other Technical Improvements
Various bug fixes.
Android Studio and the Gradle build system are now fully supported.