[android] How to run (not only install) an android application using .apk file?

You can't install and run in one go - but you can certainly use adb to start your already installed application. Use adb shell am start to fire an intent - you will need to use the correct intent for your application though. A couple of examples:

adb shell am start -a android.intent.action.MAIN -n com.android.settings/.Settings 

will launch Settings, and

adb shell am start -a android.intent.action.MAIN -n com.android.browser/.BrowserActivity 

will launch the Browser. If you want to point the Browser at a particular page, do this

adb shell am start -a android.intent.action.VIEW -n com.android.browser/.BrowserActivity http://www.google.co.uk

If you don't know the name of the activities in the APK, then do this

aapt d xmltree <path to apk> AndroidManifest.xml

the output content will includes a section like this:

   E: activity (line=32)
    A: android:theme(0x01010000)=@0x7f080000
    A: android:label(0x01010001)=@0x7f070000
    A: android:name(0x01010003)="com.anonymous.MainWindow"
    A: android:launchMode(0x0101001d)=(type 0x10)0x3
    A: android:screenOrientation(0x0101001e)=(type 0x10)0x1
    A: android:configChanges(0x0101001f)=(type 0x11)0x80
    E: intent-filter (line=33)
      E: action (line=34)
        A: android:name(0x01010003)="android.intent.action.MAIN"
        XE: (line=34)

That tells you the name of the main activity (MainWindow), and you can now run

adb shell am start -a android.intent.action.MAIN -n com.anonymous/.MainWindow