cashilysh:
I wanted to make an ambient backlight project for my monitor with programmable LED strips, but I've ran into a really irritating problem:
Just got to realise that this 'weird' and 'irritating' problem is caused by yourself. The issue doesn't appear to be due to any of the hardware modules you have. It's just something or some things that you haven't done properly - that's the likely issue. Probably better to just say that you're trying to do some fault-finding for your project.
First, turn off the power to your whole system. And then get your multimeter, and put it in electrical-continuity mode, and then see whether the ground of your arduino and the ground voltage supply and the ground of your LED strip are all electrically connected together.
Also, I do agree that the soldering work can be improved significantly. Using better soldering techniques can cut down on solder joint issues.
An accurate and adequately detailed circuit diagram will definitely help to home-in on the issue.