'ON' this area or 'IN' this area?
For example: 'The family lived IN/ON this area for four years'.
I would say 'in' but I want to make sure if I am not mixing my native language here ;)

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With the word 'area' it is definitely 'in'. 
There are a few words that you could replace 'area' with, that would make it 'on' instead, such as 'block' and 'property'.

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"In" is correct here.

Here are a few "in" versus "on" examples: "We live in this city", "We live on a farm", "We live on potatoes", and "We live in poverty"

As a native English speaker, I'm not sure of the exact rules, but I think "on" is used when there is a dependence (a person depends upon potatoes) and "in" is used when the subject is "part of" something bigger (a person is part of a city).

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Good question, Adam. Would you like to chat in english?

لا تفوّت فرصة تعلّم لغة جديدة وأنت مرتاح في منزلك. تصفّح مجموعتنا المختارة من مدرّسي اللغات ذوي الخبرة وسجّل في درسك الأول الآن!