Can the phrase 'Are you through?' be used in various situations such as meals, studying, and working? And how different is this phrase from 'Did you finish ...?'
Are you through?
The phrase could be used to ask if the person has finished the task that is being done at the time of or just before(not long ago) asking. For example I'm stuck with a bug in my program and trying to debug it. My friend knows about it and wants to acknowledge my current status. He may come to me and ask if I'm through.
Did you finish?
This phrase represent the past tense and the task could have taken place long ago. For example, I'm telling my friend that I was working on a script that does something. But my friend stops me and says "Skip to the end. Did you finish it?". In this case, my friend knows that I'm no longer working on the script. If I were, he would have asked "Have you finished the script?"
A more closer phrase to "Are you through?" is "Are you done?". It can be used in all the cases that you've mentioned. For example:
Are you done with your meal/studies/work/etc?
Are you through?
The word "through" is an adjective in the above sentence. The sentence presented can be used in different situations to ask somebody if they have finished doing or using something, as follows:
Are you through with your homework? = Have you finished your homework?
Are you through with reading the newspaper?
Are you through with your meal?
It can also be used when you have ended a relationship with somebody.
Are you through with your wife?
You can also use it to ask somebody if they have achieved success in an examination or completition, for example:
Are you through to your exams?