So I2C required a bit more work to be completed, but I finally finished it. After doing so, I started playing with the NuttX OLED driver for the display I have. It seems this is some chinese alternative which is not exactly the same as SSD1306 so it required a few changes to the driver. Luckily, thanks to the awesome work in NuttX I had the display running in a matter of hours.
So, behold the proof of success:
Behind all that mess of cables you can see the Teensy 3.1 speaking to the OLED driver via I2C and running the "nxhello" example app from NuttX. In reality the text is quite crisp, but it is difficult to capture with a camera since the light is quite strong!
So, now next step is to create a custom app to start drawing custom things to the screen. I will probably display current time as a first test.
I must say that the update rate is quite slow using I2C, sadly the SPI capable displays were out of stock (actually, I bought the remaining two for the projects we have at work). Anyway, I don't think I will need a high update rate since while riding the bicycle, the eyes are not much time on anything else but the road.
I will probably start looking into the remaining components, such as the battery and charger. I think the teensy regular will be enough, so I don't need that for now. Maybe even some buttons to interact with the screen.