"My family loves me."
You're right. 家人 (jiā rén) refers more to family members rather than just family in general, it even directly translates to English as "family people". 家庭 (jiā tíng) is more commonly used when refering to family in general. Even just 家 would be fine too but it's more commonly refered to as "home/house".
我的家。= My home. 我的家人。= My family members. 我的家庭。= My family.
But hey, Duolingo's Chinese lessons aren't entirely acurate since they're new lol... If you want to look up some definitions for Chinese characters, I recommend archchinese.com (what I used for my high school Mandarin classes, it's helpful for learning how to write Chinese characters) and wordreference.com (I found out about this website when I switched to Spanish in school lol... it's actually pretty useful as an online dictionary for any language and I trust it more than Google Translate).
I also have a pdf of my high school's Mandarin textbook if you wanna read through that lol... https://drive.google.com/file/d/0ByUq645I-K69aENSeTZicnE3SGs/view
No, that doesn't make sense. "口人" on its own isn't really a set phrase. "口" is just a measure word (for things with mouths), and "人" is person(s).
- 你的家有几口人？ — How many people are there in your family? (Literally, "Your family has how many people" — or, even more literally, "...how many mouths of people" — with the implication that there are mouths to feed.)
Right, and you could also use 个ge as a measure word for people or as a generic measure word. "你的家有几个人?“ 家人 means family members, so you could say "你有什么家人？“ basically, "What family members do you have?" 口 is a measure word, so it goes with numbers or with 几, like in PeaceJoyPancake's example, or an answer, "我的家有六口人。“