I agree.

Yes. The original sentence sounds more like: John has his own child or Pete has his own. However, both statements are equivalent because

.

This is because is essentially a big disjunction , and disjunction is commutative and associative. In contrast,

because is a big conjunction, and conjunction and disjunction distribute over each other in a more complicated way.

In the first formula, should be . Again, these formulas are equivalent because

providedy is not free in A. Using this fact, you can pull both existential quantifiers up front.