Hello, Ahmed (by the way, sorry for not having realized your name contained spaces last time I wrote to you)
Your question is difficult. Synonyms share some concepts but are not identical, so there is always some nuance that is present in some word but absent in others. I'll make a try.
- real or true news/ real or true friends
"real" can be used as a mere intensifier, as in "real friends." "true" sounds to me more like "faithful" or "loyal" in the phrase "true friends."
If you say some news is real or true, you mean it's not fake or false, so in this case they are close synonyms.
- I’ve heard that the ancient Egyptians had illnesses that were similar to ours today. Is that correct / true?
"correct" is the opposite of "wrong," while "true" is the opposite of "false." In this case, both words are quite similar for practical purposes.
- What proof do they have that this is real - true?
"reality" is opposed to "fantasy," while "truth" is opposed to "lie." Something can be real and not be true, I guess.
- And is it true / right that they did not eat much meat?
This is the sentence where I find the greatest difference in meaning. With "right," the question is not asking about the truth but about the convenience of their not including much meat in their diet (similar to: Was it good for them not to eat much meat?)