Adalight Please help a novice!

OK I don't really get it. I'm a total novice. Please don't hesitate from answering because of this. But I made the 'mistake' of buying some WS2812B LEDs in a strip. In fact the strip I bought can be found here: http://www.ebay.co.uk/itm/301594683845?_trksid=p2057872.m2749.l2649&var=600482212270&ssPageName=STRK%3AMEBIDX%3AIT

I bought an Arduino Uno from here: http://www.ebay.co.uk/itm/171518348792?_trksid=p2057872.m2749.l2649&ssPageName=STRK%3AMEBIDX%3AIT a 5V 2A power supply from here: http://www.ebay.co.uk/itm/171279744317?_trksid=p2057872.m2749.l2649&ssPageName=STRK%3AMEBIDX%3AIT

And a nice perspex box to put it in.

However this is the point where I get lost.

I have tried to follow the tutorial from here: [COMPLETED with TUTORIAL] Ambilight - Atmospheric lighting behind your TV/Monitor - Home Theater Equipment - Linus Tech Tips

and have become a little confused by this image: http://i.imgur.com/4GjqN8e.png

This appears to show that the blue - and red + from the power supply are connected to the LEDs themselves and not to the Arduino? Is this correct? Also what is the little green connector 'thing' that's connected to the power supply and where can I get one of these? Presumably it's purpose is to make connecting the LEDs to the power supply simpler without the need to strip and solder cables together.

I read this forum post: https://forums.adafruit.com/viewtopic.php?f=47&t=49537 which said I could use the WS2812B LEDs I bought and that I should 'use' the FastLED scrip specifically mentioned in this post on that thread: Adalight and WS2812B - adafruit industries

However the one significant difference is that I'm using an Uno and he's using an Arduino Nano (I think?) Is this code compatible with an Uno, or if not, is there any way to make it compatible?

Also what software exactly do I need? I'm sure I can work out from tutorials how to upload software to the Uno, but is the FastLED script all I need, or do I need to upload, or do anything else?

Also I bought these strip to strip connectors to give me some placement flexibility: http://www.ebay.co.uk/itm/180896644981?_trksid=p2057872.m2749.l2649&var=480153210401&ssPageName=STRK%3AMEBIDX%3AIT

Are these compatible? Will they work? Do the LEDs need to be stripped of any protective plastic coating in any way to make them work?

Any help would be much appreciated.

Also I bought these strip to strip connectors to give me some placement flexibility: http://www.ebay.co.uk/itm/180896644981?_trksid=p2057872.m2749.l2649&var=480153210401&ssPageName=STRK%3AMEBIDX%3AIT

Are these compatible?

No

Will they work?

No

Do the LEDs need to be stripped of any protective plastic coating in any way to make them work?

No

There are several different types of LED strip. You have to have the right hardware and software to make them work. When I say right I mean software that works with the type of LED you have.

I have tried to follow the tutorial from here: http://linustechtips.com/main/topic/120078-completed-with-tutorial-ambilight-atmospheric-lighting-behind-your-tvmonitor/

That tutorial is for a different sort of LED strip than you have. So as you can't yet change that software you will have to find another tutorial.

However the one significant difference is that I'm using an Uno and he's using an Arduino Nano

Makes no odds.

Look at this series of web pages

This uses the sort of LED strip that you have.

Also what software exactly do I need? I'm sure I can work out from tutorials how to upload software to the Uno, but is the FastLED script all I need, or do I need to upload, or do anything else?

You need a library to get your Arduino to be able to talk to your LED strip and then you need software in the Arduino to talk to the library.