It feel like it's fairly common to just say "我家人" without the "的." When I first started living in Taiwan, I was told I overused "的." Is it a cultural difference or is this program just being really strict?

Both are acceptable, also when there are multiple 的 in a single clause, usually all but the last are omitted

It's just a bit strict. Informally, you can leave out 的 between stuff like 我家人 or 我哥哥.

All these people are correct. Minimise de usage. Especially for family and friends.

Still needs some tweaking - "likes to eat FRUIT" Another acceptable translation would be "My family very much likes to eat fruit." if you haven't gotten that one already, a bit closer to the Chinese sentence here. Otherwise "....REALLY likes....." might be something similar to "我家人真实很喜欢吃水果。" Correct me if my Chinese is wrong - You'd certainly hear this in colloquial speech, but I'm not too sure if it's entirely correct from a more strict grammatical viewpoint.

would 非常 be suitable in this sentence？(我的家人非常吃水果)