Could this driver https://www.elfa.se/elfa3~se_sv/elfa/init.do?item=73-296-26&toc=19777 be an alternative to the MOSFET? Any gains by doing that?
No,
The UCD7231 high current driver is specifically designed for digitally-controlled,
point-of-load, synchronous buck switching power supplies.
It would complicated things far above your comprehension.
Ok 
What kind of driver chip do I need between the Arduino and the MOSFET? Mentioned by Pito
If you get a logic level FET then all you need is a 100R resistor to limit the current from the arduino pin. There is no need for a driver at these sorts of currents.
When you do need a driver there are lots available see:-
The power FETs and resistors are now in place and it runs perfect. The LED-stripe is working just as it should. Thank you for all help.
For you who intend to use MOSFETs I can recommend this shield i found: http://www.robotshop.com/eu/productinfo.aspx?pc=RB-Spa-505&lang=en-US
I´ve ordered one of those for "integration" of the FETs to the Arduino.
I also bought a Cupperhead WiFi-Shield. Next step ahead.