a) You're oversimplifying the differences here. 'Great' can refer to size: the Great Hall (it might not be that fantastic, but it would be the largest), I spent a great deal of money (nothing to do with the quality and everything to do with the quantity. Also, how would you contrast this usage with 'no big deal'?). Find further examples where the words don't collocate, and examples of where they have different meanings and senses and then you will see why they cannot be taught as exact synonyms.
b) Your explanation is partially right; I wouldn't say a 'high man', but imagine how your students would react after that explanation when they opened a book and saw 'He's very high up in the company'. There's an area where the meanings diverge- high society, high positions, etc.
c) A person can be lazy or idle (with the same meaning IMO), but your computer is not lazy, but it can be idle. It can also idle (verb), just as a car engine can idle at traffic lights.
d) Can you really justify the statement that 'begin' is more formal? If so, back it up with evidence. You haven't identified meanings where only one will work. Would you start or begin an engine?
Do you know this site: [DAVIES/BYU] British National Corpus ?
Enter words and phrases and get real examples from the British National Corpus. So, try entering these words, and look at the list produced and see if you can switch the word you have entered for the other term. This way, you will start to see the contexts and subtleties that determine our choices of word. Outside the fairly narrow confines of most dictionary definitions, words can have so many different uses and shades of meaning and it is in contrasting these that you will see the answer to the basic question, which is that you should not teach them as exact synonyms because they plainly are not.