Q: email does not work after Two-Step authentication feature turn on

Background:

I set up my XXX@me.com email account on my Andriod device, getting email normally. Works fine.

Issues:

I just turned on Two-Step Authentication today, and my email account logged out automatically, and can never login in again.

I tried to delete the account, re-setup, but doesn't work, says verification failed.

Result:

I have to turn off Two-Step Authentication, then re-setup my XXX@me.com email account on my Andriod device, works normally now.

Side issues:

During this process, my rescue email address changed by Apple website automactically, and cannot receive the rescue verify code, I have to delete that email address, but can never use that email address again, I have to get a new rescue email address. this is not good.

Advice:

I hope Apple can look at Google's Two-Step authentication, it's much more logical.

Looking forward to see the improvement. Or, if anyone can give some advice. It will be much appreciated.

Thanks

Amos