Can "that" be used when antecedents of the defining or non-defining relative clauses include both people and things? For examples: John, do you remember the woman and her dog that were killed in the park last year?
Yes. That as a relative pronoun can be used with people, animals and things in the plural in the collective sense, i.e. can be replaced by "them" or "they". It is used only with defining clauses:
"The road was filled with the people, carts and cattle that the soldiers were protecting." -> "The road was filled with them."
"The people, carts and cattle that the soldiers were protecting moved slowly." -> "They moved slowly."
This does not work with "which" in non-defining clauses, as "which" relates chiefly to the last mentioned thing/animal and "who" is generally (but not always) restricted to people.
The woman and her dog, which was called Fido, were killed.
The dog and the woman, who was called Jane, were killed.