Hello everyone,
I finally got everything to work! 
It turned out that there was a very small scratch in the GND lead on the PCB and that was what caused the voltage drop of ~1V. I put some solder on it to reconnect it and now eveything works perfectly, both with text and the timer.
@bperrybap
Thank you so much for your persistent help and for all your advice, it is highly appreciated! I will also implement the current limiting resistor on the BC517 as you suggested and then hopefully everything should work. I'll see if I can get by using D13, but if not I will have to change that around. But all those things should be minor issues. Again thank you very much.
If I can get this to work now, I will save the lĀ²C for my next project. I am definitely done with using LCDs that require 12 leads 