Edit: TLDR: 4 out of the 8 cables that were supplied with the FDTI were broken, The Orange Pi Zero RX pin is probably broken (Lamp on FDTI turns off), Ethernet problem is fixed by manually going into Armbian-Config, still not sure what is causing no ip to be assigned on boot of device.
I actually did this (Thanks great debug methode!) and found out that 4 of the 8 cables that they send me didn't make good contact with the FDTI.
So I selected the 4 working cables, which worked when connecting the RX and TX together on the FDTI (And echoed the input).
And I think I figured out part of the issue:
The RX pin of the Orange Pi Zero is broken, because I borrowed the same Orange Pi Zero from a friend of mine and with the same SD card, Power Cable and Setup it did work[ (Including login in as root).
Log of Orange Pi Zero without working RX pin: (The send lamp on the FDTI stops working once I connect the cable to both ends).
Log of working Orange Pi Zero:
So now about the Ethernet problems, the log of the working Orange Pi Zero:
However, when I went to armbian-config -> Network -> eth0 -> IP -> DHCP, it got an IP and the internet is working now on the device.