I am using the uptodate android app of JA Sensei.
I think I found two problems with the quiz of lesson 9 'At the clothes shop'.
1. There are question about the chronical order of 'ima', 'kinou' and 'ashita' which belong probably to a different lesson. 'Ima' appears in this lesson, but i think the other two haven't appeared up to this point in the lessons. After looking through the vocabulary section i saw that both appear in the vocabulary list of lesson 18 'How is the weather'.
2. There is the question 'To translate "to wear a hat", you use the verb: A) Kiru, B) Haku, C) Kaburu '. 'Kiru' and 'haku' are introduced in this lesson. But i am very sure 'kaburu' doesn't appear before the test. You could say that you can rule out 'kiru' and 'haku' because they are used for other kinds of clothing but i am personally not a fan of introducing new words this way.

1. Actually, regarding kinou and ashita I had the question also and in both tests. But these are basic questions I think and they do not really need to be tied to the theme of the lesson.

2. Indeed, there is a difference in the use of verbs to "put on, wear" clothes. It is best to learn them off by heart.
Japanese has at least six different verbs that mean "to wear/put on". We use different expressions depending on what you wear.

So here they are:

かぶる (kaburu) - hat (i.e. something you wear on your head)
着る (kiru) - jacket, shirt, blouse, dress, suit etc. (i.e. tops and all-over clothes)
履く (haku) - pants, skirts, shoes (i.e. bottoms, including shoes)
かける (kakeru) - glasses, sunglasses
する (suru) - watch, necklace, earrings, ring, belt, necktie, makeup, etc. (i.e. accessories and makeup)
つける (tsukeru) - perfume, cologne

What happens if you use a wrong verb? For example, if you use the verb kaburu to mean "to put on a pair of pants" (that'd be パンツをかぶる Pantsu o kaburu), it will mean that you put on a pair of pants on your head. Very funny. And if you say パンツを着る Pantsu o kiru, it will mean you wear pants around your upper body somehow. I can't quite picture how you manage to do that though!