So questions;
Is there a different version of the driver chip for common anode/common cathode displays?
The ht chip can work with either type, as long as you have no more than 8 cathodes and 16 anodes. In your case, you will be "commoning up" your cathodes/segments, in other words, red segment A, green segment A and blue segment A will all be connected to the same "com" pin on the chip. You will also have 9 anodes/digit connections to 9 of the 16 "row" pins on the chip.
You won't need any series resistors, that's all taken care of by the chip.
Best of all, only 2 wires going back to the Mega (plus 5V & ground of course) but these two wires must go to the SCL and SDA pins on the Mega. This chip uses the "i2c bus" to communicate with the Mega.