When you develop smart contracts using the remix, we got an address for that contract, I wanted to know does the contracts are permanently available to be used? or they are deleted after the certain time?
Remix works on Ropsten testnet by default (Run => Environment => Injected Web3 Ropsten), so the contracts deployed will exist on testnet as long as testnet keeps alive (months or years, depending on Ethereum).
But if you deploy your contracts using the sandbox environment JS VM, they exist only on the memory of your browser, so they don't belong to any persistent blockchain.
Yes deployed smart contract are permanently store on ethereum network if your selected environment is livenet/Mainnet in remix.
Remix Injected provider works on your selected network type on web3 it can be Ropsten testnet or Main net or others, which network you select on the web3, remix by default set it as Environment.
you can also set your own Web3 Provider