Why can't I trigger my relay?

This may be the answer, as I don't see anything else wrong in the schematic, and your other tests point to the transistors' not switching properly: according to its datasheet, pin 3 of the MMBT3904 is the collector, and pin 2 is the emitter; in your schematic, they have been swapped.

You may want to measure the voltages between the collector and ground when you toggle the pin to see if the transistors are behaving as they should and are connected properly. The measured voltages should be about 5V and about 0.3V.