How to bypass the username as a required field in registration and just use email address instead?

The registration and login forms can be customized easily below a diggest from the codex:

Customizing The Registration Form

Theme and plugin developers can customize WordPress’s built-in user
registration page through the use of hooks.

Customizing the registration form involves utilizing the following
three hooks:

1. register_form
   Allows rendering of new HTML form elements.
2. registration_errors
   Perform validation on form registration fields.
3. user_register
   Save custom form data.

Customizing the Login Form

You can customize your WordPress login page with action hooks and
filter hooks, including:

• Actions in the of the document: login_enqueue_scripts, login_head.
• Filters in the : login_headerurl, login_headertitle, login_message, login_errors.
• Actions at the bottom of and below the form: login_form, login_footer.

To additionally inform yourself about the above mentioned hooks take a look at the codex page about the Plugin API, you will find the hooks at the Action Reference and the Filter Reference. Some additional, not mentioned hooks can be found at Adminstrative Actions Section especially. Not all of the hooks are documented well (or at all), so it might be necessary to take a look at the source.