"Our" is the determiner used by the speaker and, in your example, it includes a reference to one (his father) or more other people. He refers to his father and, in particular, his father's cooking. "Come have lunch at 'our' place."

The speaker clearly lives with other people because she mentions her father cooking. So she lives with at least one other. It would sound quite strange on English to say 'my place' about a place you share.

Both ‘my’ and ‘our’ are natural in the US. An adult would be more likely to use ‘our’ especially if other residents of the house are there. But two children talking could easily say ‘my’. For example: Let’s play at my house today. I want to show you what I got for Christmas. You can eat at my house. My sister will make us something good.

Because the person talking is a child. She does not own the house; there are more people living there. “Our” here means her and her dads house

Regarding the question you asked, Terry and Shevelen8 are correct. It's Natalie's family's place, therefore 'our place' is right. The rest of your answers are correct.