What rakat in sujud do you make a dua at, can you also do it after tashahhud? And can you make a dua from your heart or from the quran?
There's no specific way to do so nor any limitation on what you may say or not. It is always recommended to use du'a that is approved (from the qur'an and sunnah). But saying a du'a (supplication) that is of your own words and expresses your feelings, needs or seems to suit you in the situation best is fine.
Note that you may say it after before or instead of the known dhikr of sujud it appears in Hisn al-Muslim actually as du'a as-Sujood:
Subhana Rabbial-'Ala (Glory be to my Lord Most High) سُبْحَانَ رَبِّيَ الأَعْلَ
What you must know is that you say the du'a in the state of sujud (prostration). And during the preayer (salah) you must follow the rulings and movements of it: anything that is not part of the prayer may make your prayer invalid.
All this is in alignment with the Sahih hadith:
The Messenger of Allah (ﷺ) said: The nearest a servant comes to his Lord is when he is prostrating himself, so make supplication (in this state).