Normally, o requires the locative case. When does it requirse the accusative? "o rodzinę"
Good question. It seems that 'pytać' somehow overwrites the 'o' = Locative rule, and it needs Accusative, both with the person you're asking (go) and about what (jego rodzinę).
Anyway, "o" requires Locative mostly when it means "about", so this is still a bit of an exception. Maybe there are more like that, it's hard for me to say. But there are contexts that are no exceptions but just need Accusative as a rule, have a look here.
No. Imperative will probably take the perfective "zapytać", so zapytaj (singular you), zapytajcie (plural you).
Probably, because pytaj/pytajcie would be possible, but that would be a request to ask him regularly, e.g. ask him everyday.
There's a proverb: "Pytajcie, a znajdziecie" = Ask, and ye shall find.
No. Firstly, "swojego rodzinę" is just grammatically wrong, because "swojego" is masculine.
Secondly, the grammatically correct "Pytasz go o swoją rodzinę." has a completely different meaning. As forms of "swój" refer to the subject of the sentence, that would mean "You are asking him about YOUR family".