«Н» gets prepended if the preposition is the main word after the pronoun. However, here preposition modifies «отце», and «отце» is the main word, so you don’t add н-.
Usually, о is used before the consonants (and е really starts with a consonant Y here) and об is used before vowels. However, there're a few words that use not о but обо: обо мне 'about me', обо всём 'about everything'.
However, in a few phrases (such as «биться о(б) стену» 'to bang [one's head] against the wall), об might be used before consonants.
In speech we treat the prepositions the same way as if they were parts of the following word. So believe it or not, the "robot lady" says it's correctly. We pronounce "о" as "o" when taking it out of context, but "о его" we would treat as a single word "оего", with the accent on the second "o" and reduce the first one to "a".
«Про» is sligtly more colloquial. For example, in names of laws, you only see «о», but not «про», because it's too colloquial for legal documents. In most cases you can use them interchangeably, they mean the same thing.
Note that the noun should be in different forms after these prepositions. «О» («об» before vowels, «обо» before «мне» and «всех») requires prepositional case: «о его отце». «Про» requires genitive case: «про его отца».
It's a preposition, like "about". Here, it is used to introduce a topic about which you don't know.