Is it permissible to ask dua in state of sujood outside salat? i.e. just for sake of DUA, can we do sujood?
You can make as much du'a as you like in sujud in any language outside of prayer. The only exception is that you can't say the Quran while in sujud.
The Prophet said [nearest translation] = إِنِّي نُهِيتُ أَنْ أَقْرَأَ رَاكِعًا أَوْ سَاجِدًا فَأَمَّا الرُّكُوعُ فَعَظِّمُوا فِيهِ الرَّبَّ وَأَمَّا السُّجُودُ فَاجْتَهِدُوا فِي الدُّعَاءِ قَمِنٌ أَنْ يُسْتَجَابَ لَكُمْ "I have been forbidden from reciting the Quran when bowing or prostrating. As for bowing, glorify the Lord therein, and as for prostration, strive hard in supplication for it is more deserving of a response."
There are many duas mentioned in hadith when Prophet Muhammad (PBUH) was in sajood. Sajda is the best time for duas, as you are closest to Allah at that time. So it is allowed to ask any dua during sajood.
Prophet Muhammad (PBUH) used to take quran as a "living book". When he recited it, he used to respond to the words. If verses come, that shows Allah's wrath, he used to seek His refuge and verses comes where His praise was done, he also used to go in sajood. So there is no limitation of when you can do sajood.
Combining these two, you can make sajood when you feel like it and ask Allah anything you like in it.
We should make much dua' in the sujood of Salat, and to make sujood for Dua' outside of it is a Bid'ah unknown to the Prophet (peace and blessings be upon him), his companions and the Tabi'een.
أَقْرَبُ مَا يَكُونُ الْعَبْدُ مِنْ رَبِّهِ وَهُوَ سَاجِدٌ فَأَكْثِرُوا الدُّعَاءَ
The nearest a servant comes to his Lord is when he is prostrating himself, so make supplication (in this state)
Source: Sahih Muslim
Sujud is permissible in 3 acts of worship:
- In the prayer it even is a necessary act. and where doing du'a is strongly empahsized (Se for example in here and here in Sahih Muslim).
- Sujud al-Quran if one recites a verse with a sajdah or listen to a reciter doing it. See for example Quran ayat of sajdah or sujūd during salat.
- Sujud a-shokr: For thanking Allah (some scholars prefer praying two raka' instead and don't accept a single sujud). For details refer to Can I perform sujud shukr without wudu/ablution?
And making dua' in sujud is sunnah based on the ahadith linked above.