Do you think it's possible Bitcoin will become a "global unit of account" in the future? If not, why?
Yes, It is possible for bitcoin to become a global unit of account in the future if it can become stable. If bitcoin can become stable, there would not be a need to check coin trackers for the current valuation as the price is not volatile, if you bought a coffee with a certain amount of bitcoin today, you will expect to buy it with the exact same amount the next week.
What would a world without Fiat really look like?
In my opinion, it cannot function. In a completely utopian society, maybe it will be possible, but such does not exist.