Is it permissible to speak in the bathroom or lavatory? And is it obligatory to close the bathroom door and turn on its light? And what is the extent of the validity of that?

Speaking in the bathroom has two situations:

1. While relieving oneself: Speaking is disliked in general, except out of necessity. 2. When not relieving oneself: Speaking the name of Allah is disliked out of reverence for Him, while other speech is permissible.

As for closing the door, it is obligatory to cover one's awrah, and it is sunnah even if there is no one who can see. However, closing the door for the sake of warding off devils or lighting the toilet to protect oneself from jinn has no basis; rather, it is sufficient to mention Allah upon entering and to seek refuge from male and female evil spirits (al-khubuth wal-khabaa'ith).