Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
1) There are no clear evidences on this subject. However, Du'a after Tarawih comes in the ruling of Du'a after Salah. Therefore, to make Du'a individually is permissible. (Ahsanul Fataawa Vol.3 Pg.519)
Also, to make Du'a collectively will be permissible if you don't make this act as compulsory.
Please read more about Du'a performed collectively: http://www.muftisays.com/qa.php?viewpage=viewQA&question=3659
2) After every four Rak'ats it is Mustahab to sit a while and take a rest quietly or recite the Quran or Tasbeeh in a low voice or pray Nafl Salah separately during this period of rest.
(Nurul Idhah Pg.97, Durre Mukhtar vol.2 pg.46)
And Only Allah Ta'ala Knows Best.