A) 8, 10)
たくさん持っていない or よくしない are not wrong, but あまり持っていない or あまりしない are more natural in negative sentences. When you use たくさん or よく in negative, たくさんは持っていない or よくはしない sound more natural.
B) 3)
は works as the contrastive marker here.
5)
が shows 冬 is the correct answer in examples, and は could work as the contrastive marker. To avoid these misuses of は and が, the best way is to omit the subject (and that's more common in Japanese).
6)
The same as A8/10.
VII) A) 4)
Your understanding is correct.
VIII) B) 3)
(The name of a program)を見ました。 is more common as an answer since the program name is usually obvious for you who watched it.
6)
はず is not used for a future plan the speaker can control (mostly volitional verbs). You need to use つもり or 予定 in this case instead of はず.
Incidentally, はず works well for the speaker's unexecuted past plan, e.g. 先週末、日本語を勉強するはずでした (implying "but actually I didn't").