If you are doing the same pattens over-and-over, it would probably be MUCH easier to program the patterns into the Arduino, than to build the DMX hardware & software into the Arduino.
I've never done anything with DMX, but we use some RS-485 (which is the same basic hardware & electrical connections) where I work.
The DMX hardware is fairly simple. It basically just "conditions" the signal for the correct differential voltage, and it uses standard XLR "microphone" cables & connectors. (You can look into RS-485 standards & driver chips).
The DMX software/protocol takes a bit more work. For me, this would be the trickiest part of the project. I don't think I've seen any software (sketch) examples for DMX devices... only DMX controllers. If you can't find any device examples, you might have to study the controller software and the DMX protocol, and figure it out. (You'll probably need to study the DMX protocol in any case.)
You can't drive 1 meter of LED strip directly from the Arduino. You'll need a MOSFET (and separate power supply) to boost the output current. That's easy and once you figure-out one, you can make 9 more.
Dimming of LEDs is done with PWM. The Arduino Mega has 15 PWM outputs (which can be "boosted" with a MOSFET). Or, there are various LED driver chips which have PWM built-in. Most of those driver chips have a serial-signal input, so you can drive many LEDs with only a few Arduino output pins, and you could use an Arduion Uno. (You would still need to boost the output with a MOSFET.) It would be a LOT easier to use the Mega than to add a chip (or two) and figure-out and program for the driver chip's serial format/protocol.