need to use I2C to control the display to avoid controlling every digit/LED separately - am I missing anything?
There are many ways to connect a driver and extra pins. I2C is just one option you have. There is also SPI and shift registers. SPI offer many alternative just like the I2C.
As I need a supply of actually 8.2V for the Segments + V(current limiting resistor), I somehow need to boost the voltage.
I would advise against this, as voltage boost for those cheap boards are limited in current. You also don’t need anything like as precise as 8.2V, any voltage will do, just regulate the current with your choice of seriese resistor.
Best to use 9 or 12V and regulate down from that to power your Arduino and the rest of your circuit.
Accoding to the HT16K33 data sheet, the maximum voltage output seems to be VDD - max 5.5V.
If true, and I haven’t checked then you can’t use that chip for your project.
Edit - just checked the data sheet, not only is that chip not capable of supplying enough voltage it is way over the top in that you don’t need multiplexing and a key input from what I understand you want to do.