What is the ruling on using Twitter and placing a personal profile picture there, and what is the Islamic etiquette for that?

The default ruling is that it is permissible to use social media sites, except for those who use them for something forbidden or that leads them to something prohibited. The default ruling for things is permissibility, and whatever there is no evidence to prohibit it, then it is permissible. The preponderant opinion regarding photographic images of living beings is that they are permissible. Therefore, there is no harm in placing a picture of a shaykh or anyone else in the profile picture, as long as the owner does not object to it. There is also no harm in placing background pictures of people bowing or prostrating, or pictures with the name of Allah or Qur'anic verses written on them, provided that the picture is free from prohibitions such as displaying 'awrat (private parts) or ridicule.