'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?

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