Addresses never expire. Even if you generate hundreds or thousands of new addresses, money sent to an old address in future years will still be under your control, assuming you make backups of your private keys using your wallet’s backup feature and don’t keep everything on one device.
Normally, wallets play no part in receiving money but will fetch blockchain data to calculate your current balance. Your wallet is your window onto the blockchain.
You can get the transaction-ID from the exchange and check the status of that transaction-ID at any blockchain explorer.
