The new transistors have finally arrived and after soldering one full adder on the board the output leds finally lit up and everything worked as expected.

Now I just had to desolder the trasistors I've already put on and then solder the rest of the board.
Thanks to my workplace, desoldering the transistors was a breeze with a metcal desoldering station powered by shop air. After that it was just a rinse and repeat for over 700 soldering points. 
But all this soldering was quite enjoyable and worth it to have a functioning board that looks stunning. 
After all that it's time for a case. I already have a really nice peace of wood which I want to mill out as well as a CAD-Design. The only missing piece is a CNC-Mill. 
I also only now realized, that it would be nice if this board was battery operated. I will think of some way to add in a Li-Ion battery and charger circuit into the case.

In the meantime I am thinking about the next project. I wanna do something more complex. 
My Ideas so far are:
- switch to ICs to design a more complex circuit getting closer to an actual computer
- switch to SMD transistors and make a proper calculator
- make modular PCBs that can be connected  in different ways to create different kinds of computing devices/functions
If you would like to see any of the above feel free to drop me a comment, follow this project and my account.
Thanks again to PCBWay for providing the stunning PCBs for this project.