Trivia quizzes do not allow for deep chemistry nor deep philosophy. I actually sympathize more with your answers than the ones that were marked correct.
Purity is a matter of convenience and degree. No sample of water has ever been prepared that did not have sodium ions present in femtomolar (10–15 M) quantities, and at least a dozen other ions present in more than attomolar (10–18 M) quantities.
And even as an ideal concept, pure water cannot be a pure substance. The definition of a pure substance involves a collection of identical molecular entities, or a perfectly regular network-bonded lattice. But ideally pure water contains 10–7 M hydrogen ions and a similar concentration of hydroxyl ions.
"Elements occur only in the form of atoms" is, on the other hand, largely true. It really depends on whether you regard an ion of an element as an atom of that element, because elements are frequently present in the form of ions.
But the problem with the way this question would normally be interpreted is that chemists have adopted the habit of a loose use of the word "element" when they really mean "simple substance".
Dioxygen gas is a simple substance which contains only the element oxygen, as also does the simple substance ozone. Carbon dioxide is a compound which also contains the element oxygen. It does not contain dioxygen gas nor ozone. Neither dioxygen gas nor ozone is "a form of" the element ozone. But an alpha particle, He2+ is a form of the element helium, even though it is not a helium atom.