- what happens after 21 million bitcoins have been mined?
The mining process continues, but miners won't be rewarded with the usual block rewards. They will have to manage with the transaction fees only.
- What will miners do after the total of 21 million bitcoins have been mined?
Same answer as above.
- Will the total supply be increased because some bitcoin holders have died and others have lost their private keys?
Bitcoin is only Bitcoin if it sticks to the original rules or modified rules that don't negatively affect the 3 pillars of Bitcoin (transparency, decentralization, and immutability). The original rules state a total cap of around 21 million units. If you change this to 25, 50, or an unlimited cap, that's no longer the cryptocurrency people agreed to use.