Answers to those questions are not yet known. The price competition between different cryptocurrencies should be the necessary market mechanism. Someone has to pay the price of mining, and if mining is not necessary, then it's not going to be paid. If it can be proven that the same security can be achieved with less mining, then either the Bitcoin protocol is updated, or a Bitcoin fork will become more valuable (and that fork will be named Bitcoin).

There are lots of cryptocurrencies which have infinite supply, which means that the mining rewards will keep the network running. For example Ethereum, Monero, Dogecoin and Grin. Bitcoin will rely on transaction costs, but it doesn't provide fully deterministic security as far as I know.