Is it permissible to divide the Tahajjud prayer into four rak'ahs after 'Isha' and the fardh sunnah, then sleep, then pray another four rak'ahs half an hour before Fajr, then the Witr with three rak'ahs, making a total of 11 rak'ahs?

There is no objection to dividing the night prayer (Qiyam al-Layl) into multiple times, such as performing some after 'Isha and then some before Fajr. It is not a condition that it be performed at a single time. This is supported by the action of the Prophet, peace and blessings be upon him, as narrated by Ibn Abbas, where he prayed four rak'ahs after 'Isha, then prayed five rak'ahs later, and then prayed two rak'ahs.