"Scusi" is the (grammatically) formal address (corresponding to "Lei"), "scusa" the informal (corresponding to "tu") one. So among friends you'd probably use "scusa", for strangers "scusi". But be aware that either can sound too sloppy for any non-trivial offence. If you really did something wrong it's better to err on the polite side. You can keep "sono davvero spiacente" for ruining someone's marriage or something like that. https://www.youtube.com/watch?v=KniUNdVZvH4 :)
I believe mi dispiace and spiacente are the informal and formal for "I'm sorry" but scusa/i is much more like "excuse me" and as far as I know is used like it. In Hebrew there are also 2 words for sorry one equivalent to "excuse me" or "forgive me" or "pardon" and one is more like "sorry".
Is there a difference between when to use "davvero", "veramente" and "proprio"?
Which leads me a little off topic, but I get confused with the different meanings of "proprio". That word (when changed to match the subject) can mean either "his/her/their own [possession]" and "really/truly", correct?
Right, for example "lui ha il proprio letto" means "he has his own bed."
As for the other words that you're asking about, you could say "io sono proprio/davvero/veramente stanco," which means "I am really tired." But if you wanted to say "Really?", you'd never say "Proprio?" You'd instead say "Davvero?" or "Sul serio?"
living in italy I've found that nobody ever uses "spiacente". moreover when i used to say spiacente to people they wouldn't understand or looked at me weirdly. the two ways to say sorry are "scusa"(just a light hearted apology, like when you accidently bump into someone) and "mi dispiace"(serious apology). and use "scusi" for "excuse me", not " scusa" coz apparently thats "sorry"
davvero means "really" in the sense of "truly" or "actually". molto means "really" in the sense of "very", not questioning it's authenticity, but it's magnitude.
if you split up the word "davvero", you almost have "da vero", or "of real"
Thanks for asking, I never really noticed how "really" has two meanings in english. This is why I love learning languages!
It's literally the difference between really and very. We often use really (davvero), or truly, to mean very (molto), as in "I'm really tired", but sometimes really/davvero carries the original meaning of being real and not fake. (Davvero = Da + vero = in truth.) In a sentence like "I may look 15 but I'm really 21", you can't replace "really" with "very". Of course, "I'm really sorry" is ambiguous, and could either mean "I'm very sorry" or "my sorrow is sincere".
By the way, the word "very" comes from the Latin "verus", meaning "true", which is the same as the root for davvero. People have been using "truly" to mean "very" for a very long time.