Is it possible to have staking or reward functionalities, maybe minting, with your Native Tokens in Cardano?
This is not a native functionality on the Cardano blockchain. However, this will for sure be possible with a custom smart contract after the Alonzo release.
This form of "staking" would not require a staking-pool and would not produce any blocks or get rewards. It would be an independent system build by you with your own custom rules running on the Cardano blockchain.
No, that would be very bad for the platform, if any staking reward mechanisms on protocol are done based on native tokens which cost few seconds to mint and burn with no tangible outcomes, as they do not represent stake on cardano. Having said that projects can certainly devise incentives based on history of holder, but such rules will not be at protocol level, possibly only at contract level or off-chain