Does Android keep the .apk files? if so where?

You can use package manager (pm) over adb shell to list packages:

adb shell pm list packages | sort

and to display where the .apk file is:

adb shell pm path com.king.candycrushsaga
package:/data/app/com.king.candycrushsaga-1/base.apk

And adb pull to download the apk.

adb pull data/app/com.king.candycrushsaga-1/base.apk

Preinstalled applications are in /system/app folder. User installed applications are in /data/app. I guess you can't access unless you have a rooted phone. I don't have a rooted phone here but try this code out:

public class Testing extends Activity {
    private static final String TAG = "TEST";
    @Override
    public void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.main);
        File appsDir = new File("/data/app");

        String[] files = appsDir.list();

        for (int i = 0 ; i < files.length ; i++ ) {
            Log.d(TAG, "File: "+files[i]);

        }
    }

It does lists the apks in my rooted htc magic and in the emu.

If you just want to get an APK file of something you previously installed, do this:

1. Get AirDroid from Google Play
2. Access your phone using AirDroid from your PC web browser
3. Go to Apps and select the installed app
4. Click the "download" button to download the APK version of this app from your phone

You don't need to root your phone, use adb, or write anything.