majenko:
Let's start with the simple one.For the LEDs you will need resistors - 120? to 150? should be fine (R=V/I = (5 - 2.8) / 0.02).
As for the chip, all that does is convert the 0-5V of the arduino to the 0-3.3V that the display needs. You can achieve the same result in a large number of ways, not all requiring a chip.
For low-speed communications resistors arranged in "potential dividers" can be used. For higher speed you can use diodes, transistors, any number of methods.
I ended up not needing any resistors because the digital i/o pins have very controlled current so the LED did not blow or even raise in temperature... I just connected led to GND and digital pin 13 and that worked just fine. The led is 2.8 v with 20ma current... The arduino has a 3 v power supply too so I would think that works well with this led as well.