My question is;
- what happens after 21 million bitcoins have been mined?
There will never be 21 million bitcoins. The total will never exceed 20,999,999.9769
And, because some miners have failed to pay themselves the full reward that they were allowed, the total will actually be less than that.
What will happen is that the inflationary subsidy will be cut in half (truncating to the next 0.00000001 BTC) every 210,000 blocks (approximately every 4 years) until it shrinks to zero. Then there will no longer be a subsidy on the blocks.
- What will miners do after the total of 21 million bitcoins have been mined?
The exact same thing they do today. Solo miners and mining pools will continue to build blocks from transactions. Solo miners and mining pools participants will continue to perform proof of work on those blocks. Income will continue to be earned from the block reward. (Note that the block reward currently consists of the inflationary subsidy PLUS the sum of all transaction fees of all transactions in the block. Once the subsidy is gone, the reward will just be the fees).
- Will the total supply be increased because some bitcoin holders have died and others have lost their private keys?
No. The available supply in the market will be REDUCED because some bitcoin holdrs have died and lost their private keys.