MU domain mapping without using a plugin

To be honest even after your Update:

How may I fix the fact that the admin area preserves the original url
when using the Domain Mapping plugin (the one with the sunrise.php)?

I don’t quite understand whether you want to “Redirect administration pages to network’s original domain” or not.

In any case, as per the installation page:

  1. “Redirect administration pages to network’s original domain (remote
    login disabled if this redirect is disabled)” – with this checked, if
    a user visits their dashboard on a mapped domain it will redirect to
    the dashboard on the non mapped domain. If you don’t want this, remote
    login will be disabled for security reasons.

There is also an abundance of even youtube videos that may help.

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