"She has small children."
Translation:Zij heeft kleine kinderen.
No, but Klein is a very old surname of German origin that spread throughout central Europe and beyond, so I'm sure some Dutch people have this last name. His dad was Austrian, his mom Hungarian, and they were Jewish.
The notes say "If no article at all comes before a het-word, then the adjective does not get the -e ending either". Since that "Kind" is a het-word, I expected "klein kinderen" to be correct. Why isn't it? What is the correct note's section I should have read not to commit this mistake?
Ze is a very slightly informal variant of zij, just like in English, it's is a slightly informal variant of it is. You could think of ze as just a careless pronunciation of zij, but for some reason it is customary to distinguish the two in writing. It's essentially the same thing as with jij and je, except je is ambiguous. (It is not just a variant of jij, but also a variant of jou and of jouw.) As they originate in careless pronunciations, je and ze cannot be used in emphatic contexts, just like it's cannot be used in emphatic contexts. I.e., similarly to how you can't say "Yes, it's." instead of "Yes, it is.", you can't use ze and je when answering a who-question.