Hi! First of all I really want to thank you for translating it, I do not understand enough Italian to do so, and it's really impressive of you. What I'm writing here are just comments on how to make it correct and sound good in Swedish :)
"Ni har gjort mig inget" is not grammatically correct. A correct phrase would be "Ni har inte gjort mig någonting" (You haven't done anything to me)
The third line should be "Solen över Ramblan är inte den samma idag" (The sun over the Rambla is not THE same today)
The fourth line should be "I Frankrike är det konsert" (There is a concert in France) for it to sound good in Swedish.
The fifth line should be "Några sjunger kraftigT", because this is an adverb.
The sixth line means something like "It always rains in London, but today it doesn't hurt", right? If so the second part should be "Men idag gör det inte ont".
"Himlen slår inte av priset" is incoherent in Swedish, I think I'd say it like "Himlen slutar inte ta betalt" (The sky doesn't stop demanding pay).
"I Nice blev havet röTT" - hav(et) is a neuter noun, and the adjective should be in the -t form.
"Och denna enorma kropp vi heter Jord" should be "Och denna enorma kropp vi KALLAR Jorden". The verb "att heta" means to be called something, but it can only refer to a subject. For example "denna kropp heter Jorden" (this body is called Earth), makes sense, but here we are calling it Earth, and therefore it has to be "Denna enorma kropp vi kallar Jorden".
I might say "skadad i sina organ" instead of "på sina organ". Prepositions. Sigh.
"Men deT viktigaste", it's a thing, and therefore it's a det here.
"Lika som murar utan tavlor" should be "Liksom väggar utan tavlor"; there's too kinds of walls in Swedish, high indoor walls that go from floor to ceiling (vägg), which can be outdoors too, but they need to be above your head to be a "vägg", and "murar". Murar are (usually) lower and (usually) outdoors.
I might replace "Minuter av sörjande tystnad" with "Minuter av en sorgsen tystnad", but this makes sense too. However, the minutes are "brutna" (plural) as they are several minutes.
"Ni har tagit inget från mig" doesn't make sense either, same problem as in the song title. It has to be "Ni har inte tagit någonting ifrån mig" (You haven't taken anything away from me).
"För allting når utöver" does make sense, but it feels more Swedish-ish to say that "För allting annat är större" (Because everything else is bigger). It's a matter of taste though.
"Några korsar sig" --> "Några gör korstecken"
Either imamer and präster are both indefinite, or both definite, they can't be "imamer and prästerna". It's either "alla imamer och präster" or "alla imamerna och prästerna".
"Miljarder människor som hoppas på någonting"
"Ska vi byta varandras skinn, i grund och botten vi är mänskliga" --> "Ska vi byta varandras skinn? Vi är i grund och botten alla mänskliga"
"För livet är inte någon åsikt/synvinkel"
"Och det finns ingen pacifistisk bomb", not a definite noun here, so the adjective can't be definite either.
"Kolla mitt liv som fortsätter fram" --> "Detta är mitt liv som fortsätter framåt/vidare"
Replace "gångtunnlar" with "tunnelbanor"
"Kontrasts murar, byggt för bröd" --> "Murar av kontraster, byggda för bröd"
"Men emot varje skräck som står i vägen", this is not correct in Swedish "Men emot all ondska som står i vägen" (but against all the evil which stands in the way) sounds like the best way of phrasing it to me.
"Världen reser sig upp igen med ett barns le" This is a secondary clause, and we need to change the syntax for it to be correct. "le" is the verb, but "leende" is the noun. It has to be written as "Reser sig världen upp igen med ett barns leende".
After this everything that needs to be changed has been mentioned already, up until "Jag är medveten om att allting återvänder aldrig mer". It's too many evers here. I'd write it as "Jag är medveten om att allting aldrig återvänder igen".
"GlädjeN flög"; has to be a definite noun.
"Lika som en bubbla flyger bort" --> "Liksom en bubbla flyger bort".

Once again, thank you so much. I love this song, and I would never have been able to translate it myself, here I'm just trying to puzzle together it so it makes 100% grammatical sense and flows well in Swedish!