> Are cryptocurrencies supposed to be a potential replacement for real life cash?

They are supposed to be a medium of exchange. “Real life cash” is one of many forms of money; even for any particularly currency, like dollars, a very small fraction of use is “real life cash”. But, yes, in the most extreme visions, cryptocurrencies replace other currencies for all uses. More moderate visions, however, exist. So, as always when you use “supposed”, the answer is undefined without qualifying it as to by whom it is supposed.

> If so, why does it make sense that people can "generate" cash by proving some amount of work done?

Because there needs to be some mechanism to provide the currency supply, and also some incentive for people to provide the infrastructure on which the currency system relies. For fiat money systems the first is typical policy making in a central bank, and the second is government action to control competition in the banking space and to support banks, reinforcing the profitability of banks. Mining serves both of those functions in a cryptocurrency system (both reinforcing the profitability of transaction network participants and providing the mechanism by which currency supply is managed.)

> Is the main functionality of these cryptocurrencies supposed to be "people can send currency to each other", or "people generate currency -- a number -- and sell this currency for real life money"?

Participants in a currency system selling it for other currencies (FOREX) is a feature of every currency system in a world with more than one currency. Again, the degree to which each of those is “supposed” to be the main function depends on exactly whose supposition you are looking at it.