Assalaamu alaykum wa rahmatullaah. Is the following hadith authentic? The Prophet, sallallaahu ʻalayhi wa sallam, said, “Whoever performs ablution diligently and performs two rakʿahs (units of prayer) with sincerity and devotion, Paradise becomes obligatory for him.” (Sunan Abu Daawood) Also, what are the other evidences of (the two rakʿahs) of tahiyyatul wudoo'? May Allaah reward you.

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, sallallaahu ʻalayhi wa sallam, is His slave and Messenger. 

This hadeeth was cited by Abu Daawood in his Sunan, and it is an authentic hadeeth. It was also cited by Imaam Muslim in his Saheeh: It was narrated on the authority of ʻUqbah ibn ʻAamir that he said:

“We were entrusted with the task of tending the camels. On my turn, when I came back in the evening after having grazed them in the pastures, I found the Messenger of Allaah, sallallaahu ʻalayhi wa sallam, standing and addressing people. I heard these words of his: 'If any Muslim performs ablution well and then stands and prays two rakʻahs setting about them with his heart as well as his face, then Paradise will be guaranteed to him.' I said, 'What a fine thing this is!' I heard someone before me saying, 'The first was even finer.' When I cast a glance, I saw that it was ʻUmar, who said, 'I see that you have just come; earlier, he, sallallaahu ʻalayhi wa sallam, said, 'If anyone amongst you performs the ablution and completes the ablution well and then says, 'I testify that there is no deity worthy of worship but Allaah and that Muhammad is the servant of Allaah and His Messenger,' then the eight gates of Paradise will be opened for him and he may enter through whichever of them he wishes.''” [Muslim]

As for your second question, we have previously answered it in fataawa 21819 and 174127; so please refer to them.

Allaah knows best.