So it turns out that the answer was right under my nose and does not involve any deceit or hacking. Yahoo! provides an option for creating an account without a cellphone, but it isn't via the "Create Account" button.
On the sign-in page, there are a few options:
If you click the "Facebook" or "Google" button, you can sign in to Yahoo! using a Facebook or Google account. Once you sign in that way, Yahoo! creates a Yahoo! account linked to your other account, and you do not need to enter a cell phone number to create a Yahoo! account this way.
So for someone who wants to create a Yahoo! account without a cell phone, the steps are
Create a Google or Facebook account.
Sign into Yahoo! using the Google or Facebook account.
Use that sign-in to create the Yahoo! account you want.
To use this method, however, you give Yahoo the right to capture any & all information that is on your Facebook page. Be aware of this before making the connection to Facebook!
This might be security purpose they are not allowing you to access your account. They wants verify your identity. Try to reset your password.
If you don't have option of mobile number and alternate email address, you can give answer to security question to reset password.
Reset password <- use this link to direct reset your password.
Click on above link. Go to Password and sign in -> Problem with mobile code and notification -> click on Email(envelope symbol), now write your problem to Yahoo customer care and wait for their response.
Best Answer
From Yahoo! help:
See this -> To change your listed mobile number in Yahoo Messenger.