Can the EVM update deprecate operations of older smart contracts, making these contracts non-operative? If so, what are the solutions to upgrade these smart contracts to the new version of the EVM if they cannot be modified directly (due to immutability)?
Yes, and in fact some already have in the past due to changes to the gas schedule.
Designing a contract to be upgradeable like xgabrielx said is a good practice due to unforeseen changes and problems like this.
Well yes, over time some functions might be depricated (as is standard procedure, not limited to EVM).
(1) You deploy a new contract, telling users to start using that instead.
(2) You have a fronting contract which points to the "latest" version of you main contract.
(3) Use ENS to mycontract.eth and point it to the latest version of your contract.