After mining all the Bitcoins, miners start processing transactions, receiving a reward for it. The Bitcoin network will live and I will say more, it will be very successful because, unlike fiat money, Bitcoin will suppress inflation due to its limited count
That won't be the case, when all the Bitcoins were mined, there are no more rewards for every block that will be mined. Instead, the minerss will have to place higher transaction fees just to sustain the profitability of mining. With this, we can assume that transaction fees of Bitcoin will be very high compared to the current fees.