What are the precise contexts of the word usage of the aforementioned words for sorry?
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I look at http://www.wordreference.com/enit/sorry to clarify the different usage of the italian words you mention:
"I am sorry that you were sick" = "Mi dispiace che tu fossi malato".
"Sorry, but we sold out the tickets" = "Mi dispiace, i biglietti sono finiti". Please note that this example is more formal and you can as well use "Spiacente" ("Spiacente, i biglietti sono finiti")
"Excuse me, sir; may I ask you something?" = "Mi scusi, posso farle una domanda?".
"Sorry? I have not understood" = "Scusi? Non ho capito".
When used alone, "Scusi" has the same meaning of "My apologies", and can be replaced with "Mi dispiace" (but "Scusi" is more polite).