From Tafsir al-Maududi:
وَعَلَّمَ آدَمَ الْأَسْمَاءَ كُلَّهَا ثُمَّ عَرَضَهُمْ عَلَى الْمَلَائِكَةِ فَقَالَ أَنبِئُونِي بِأَسْمَاءِ هَٰؤُلَاءِ إِن كُنتُمْ صَادِقِينَ
After this he taught Adam the names of all things. Then He set these before the angels and asked, "Tell Me the names of these things, if you are right (in thinking that the appointment of a vicegerent will cause disorder)".
"Nomenclature is the means by which human mind grasps the knowledge of things. Hence, the whole information of man, in fact, consists of assigning names. for things. Thus, teaching Adam the names of all things was meant to impart their knowledge to him."
قَالَ يَا آدَمُ أَنبِئْهُم بِأَسْمَائِهِمْ فَلَمَّا أَنبَأَهُم بِأَسْمَائِهِمْ قَالَ أَلَمْ أَقُل لَّكُمْ إِنِّي أَعْلَمُ غَيْبَ السَّمَاوَاتِ وَالْأَرْضِ وَأَعْلَمُ مَا تُبْدُونَ وَمَا كُنتُمْ تَكْتُمُونَ
Then Allah said to Adam, "Tell them the names of these things." When Adam told them the names of all those things, Allah declared, "Did I not tell you that I know those truths about the Earth and the Heavens which are hidden from you? I know what you disclose and what you hide."
'This demonstration was the answer to the first doubt of the angels. Thus, Allah, so to say, informed them: "I am not giving Adam only authority but also knowledge. The chaos that you apprehended from his appointment is only one aspect of the matter. It has its good aspect also, which is more weighty and valuable than the evil aspect, and a wise man does not give up a greater good because of a lesser evil."'
1. What names is Allah (SWT) referring to? His own names? Name of the Angels? or there is any other names?
According to Maududi for ayat 2:31 the names are everything in the universe. Allah had taught Adam knowledge of everything and therefore everything would have names so the 'names' is referring to the names of everything that Adam has learnt about the universe.
However, the second use of names is when Allah tells Adam to tell the angels their names, so this is the other context where names is used.
2. Why did Allah felt the urge to confront with Angels regarding creation of prophet Adam (PBUH) ?
According to Maududi, the angels doubted Allah's action to appoint a vicegerent because they thought it would cause disorder. Allah then told the angels to name somethings, but they didn't know what it was and Adam did. Allah is saying that Adam has knowledge, so when he is appointed, although it may cause little chaos, his knowledge is greater, giving him authority. So the key lesson here is that a wise man wouldn't give up the greater good because of a lesser evil. He also says that He knows everything and the unknown doubt him.
As we know in Islam, knowledge is very important. There are many ahadith which support this.
Allah knows best.