That device supports live data, however it shows nothing when connected to my omega other than those two error codes.
Pedal trick is notably more reliable than some code readers. I'd ignore the code reader here tbh.
Did the car refuse to start In the video? It didn't sound good at the end. 
Its worth further investigation before buying parts IMO, you need a bit more info to go on. (As as Mark Dtm says.)
The car starts fine, no issues at all. It just runs like it was going to explode any minute. This actually happened when I was driving once before. It was fine after I switched the engine off and on. This time it also happened when I was driving, this is how I got home, about couple of miles, but a simple off/on didn't help and now it runs like it.
When I bought the car I don't think those coils were even replaced since the card left the factory and I certainly didn't do it. I'm the second owner.
I don't have a problem getting new parts, especially when it comes to plug business only because I know it was never replaced, so perhaps now is the time.
What else could be wrong? The MAF sensor is rather fine as I replaced it (with genuine bosch) about three years ago and I am aware what kind of behaviour to expect when it fails, so I believe the MAF is OK.
The mileage is about 95k.
Anyway, what do these codes actually mean? There's 6 codes, but 0304 missing?