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