IIRC, it is not explicitly stated if you get your clan name based off the father and mother. But typically the wife would take the husbands name, for instance Hinata Uzumaki.
If you go by the principle that the wife will take the husbands name, which does not happen on a few rare occasions, it is just common sense that you'd get the fathers name, as the mother married into the clan. However, Minato is of Namikaze and Kushina is of Uzumaki, yet Naruto is named Uzumaki. Kushina never took Minato's last name because Minato had tons of enemies that sought his life and if Naruto was named Namikaze, it'd make those same people go after Naruto. Naruto wasn't named Namikaze because that would have put him in danger.
Now, for your second question.
It is correct that all chakra started from Kaguya Otsutsuki, but that does not mean that everyone should be related to her. Remember, when Kaguya had Hagoromo and Hamura, the chakra was accidentally inherited into them. Kaguya tried taking it back which eventually lead to her being sealed by Hagoromo and Hamura. Hagoromo saw chakra as not a weapon, but a power that can link everyone's soul and hearts, leading all to love each other.
Hagoromo saw chakra as a power that would bring people together, and did so by giving others chakra that did not directly inherit it from his mother, Kaguya. Because of this, clans like the Aburame and Inuzuka are most likely descendants of those Hagoromo passed chakra onto. Remember that once Hagoromo passed chakra onto others it would now become an inherited power.
Uzumaki, Senju, Uchiha, Hyuga, and Kaguya clan are all descendants of the Otsutsuki, more specifically Kaguya, and other clans such as the Inuzuka are most likely descendants of those Hagoromo passed chakra onto.