You could just turn off Google's 2-step authentication and you will only need the password to access your google account. You will still get email when a unusual login happened. It will say something like "Someone is logging in to your google account from XXXXXXX, if this is you, you could ignore this notice..blah blah blah..." I believe you get the email on your gmail account and on your recovery email account. So even if someone actually broke into your gmail and deleted that notice, you will still see it on your recovery email account.

The other day I was trying to locate my wife's iPhone using Find my iPhone on iCloud. I came into this 2-step authentication with iCloud problem where it wants to sent a verification code to the iPhone to verify the computer I am using is legit. The problem is I don't have the phone.. and I have to use iCloud to find it...