Is the wallet ID you use in your blockchain.info wallet the same as your public address where others can send bitcoins? And the 'Backup Phrase' is the same as your private key? Maybe if someone could explain all the levels of authentication in the Blockchain.info wallet?
No, your wallet ID is only used for logging in, keep it to yourself. Your backup phrase is used to generate your private keys, but it is not the private key itself. It uses the phrase as a seed to generating a Heirarchical Deterministic wallet as per BIP 32, 39 and 44. So backing it up is equivalent to backing up all private keys in your wallet.
People send BTC to a bitcoin address, which you can generate when you log in.