Answered by Shaykh Abu Usamah
1 & 2. Du`a (i.e. Supplicating and asking of your needs from Allah Ta`ala) is permissible after Fard Salah and in fact recommended as according to the hadith this is one of the times when du`as are accepted. The du`a made at this time can be in any language and is not part of the salah.
However the du`a made in salah before salam, must be in Arabic, preferably the du`as recorded in the Qur’an and Hadith. It must not resemble the talk of people.
Saying Subhanallah, Alhamdulillah, Allahu Akbar, 33 times each and completing 100 by reciting La ila illallah (or similar forms of dhikr) known as Tasbihat or Dhikr is also recommended after every Fard salah. These are not to be done before the termination of the salah, i.e. before salam.
(Radd al-Muhtar 1.477, H.M. Saeed Publishers)