ATMEGA328p only outputs 1.15V
I finally figured out why the circuit did not work. At least I got my LED flashing, I'll have to see how it handles the relay.

The crystal was not connected to the µC, see the picture. I have no idea how this got past me as I created the schematic with the board at hand and not from the tutorial.



I am still very grateful for the inputs regarding:

Flyback Diodes

ULN2x03 (To my understanding this should also solve the problem of the output current of the atmega, I already purchased one and will use it on my next project)