1) Can we offer special supplications in the obligatory prayer at following positions: (a) After Rukoo' while standing before going to prostration (b) In the both prostrations (c) In the both Tashahuds (in case of 4 Rak'ah prayer)?
2) Also can we use our own languages other than Arabic for above supplications within prayer?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.

Supplicating when in prostration is desirable, for this please refer to Fatwa 86117. As regards supplicating after standing up from the Rukoo' position, we do not know any evidence proving it in the Sunnah. What is confirmed from the Prophet is that when he used to stand up from Rukoo', he used to say: "Rabbanaa wa lakal-Hamd (Our Lord, for You is all perfect praise)." [Al-Bukhari] The narration reported by At-Tirmithi includes the following addition: "Milas-Samaawaati wamil-al-ardhi, wamila ma shi'ta min shay'in ba'd (The heavens and the Earth and all that You will abound with Your praises)." The Prophet further said: "When the Imaam says: "Sami'a Allaahu liman hamidah" (May Allaah answer he who praises Him), you should say: "Rabbanaa wa lakal-Hamd." [Al-Bukhari]

With regard to supplicating when raising up from Rukoo', we do not know any evidence proving it, and standing in prayer is not a position that is suitable for supplicating except Qunoot supplication and the supplication for opening the payer after saying Allaahu Akbar and before starting the recitation of the Quran or when one hears a verse about Allaah's Mercy or a verse about punishment in Hellfire. Moreover, we do not know any evidence about supplicating after the first Tashahhud. Ibn Mas'ood when describing the method of the prayer of the Prophet said: "In the middle of his [Prophet's] prayer, he stands up when he finishes his first Tashahhud, but he used to supplicate when he finishes the last Tashahhud, what Allaah wills for him to supplicate, and then he concludes the prayer by saying 'Salaam'." [Ahmad and Ibn Khuzaymah]

This is evidence that the Sunnah is to supplicate after the second and last Tashahhud of the prayer and not at the first one.

As regards supplicating in a language other than Arabic, please refer to Fatwa 90022, 87759 and 81564.

Allaah Knows best.