Just to confirm that my problem is solved and that I managed to compile a working video inside APK file.
My process for functional video:
1. RexRainbow video plugin ().
2. Construct 2 Cordova export with 4.0+ Ice Cream Sandwich Crosswalk.
3. Intel XDK export/build with Crosswalk for Android.
The difference between Construct 2 Video object and RexRainbow video plugin is that video stays on top of everything else except Form Control objects and it works with URLs and not with imported video files.
I haven’t experimented much with RexRainbow video plugin, just gave it a go because I was desperate and Construct 2 video object just didn’t work. I tried it on Android browsers: only Firefox managed to play them normally. I tried with 7 different Android APK files: not one played video.
To anyone who managed to produce an Android APK with functioning video: You are the best and good luck to you!
..to the rest of us: If you need video for Android system this plugin worked for me, hopefully it will work for you too:)