1 really => I am talking about a real necessity [stronger collocation]
truly => What I say about this necessity is true.
2 'really don't mind' is a very strong collocation; if you use 'truly', you are implicitly avoiding this, so emphasizing the truth of what you say (i.e. I'm not lying')
4 really => emphasis on her: can she really have behaved like that?
truly => emphasis on person addressed: can I rely on your account?
5 'really beautiful' is a strong collocation; if you avoid it, you are implicitly addressing the issue of truthfulness again.
But I'm splitting hairs. Generally, either will do. You can also combine them, in informal contexts (often after expressing disbelief):
Did you really say that - really and truly?
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