I think we can solve the Italian problem...somewhat.

In researching Roman Legions, I came across the website livius.org. which incorporates a short history of the various armies. There were three Italic Legions, ironically named I, II & III respectively.

Italic II and Italic III were mid-late second century CE legions and apparently never saw action in Palestina (formerly Judea). Italic I was founded by Nero c.66/67CE, about thirty years after the crucifixion, so it could not have seen action in Judea at the time of Christ. However, parts of the Legion may have fought in the Bar Kochva Revolt in Judea in the 130s CE.

So if Luke wrote Acts, it is unlikely that he would have used the term Italian Legion/Cohort/ Regiment in referring to a specific army. It didn't exist for him. This leaves at least three possibilities. Acts was indeed written by Luke and later edited incorrectly. Or Acts was written later than Luke, and probably later than the Bar Kochva Revolt, by an author who could have been aware of Italic I, and anachronistically puts Cornelius in that Legion at the time of Christ. Or the Latin source quoted at Livius is wrong, in which case we're back where we started.

All of that said, this does not negate the contemporaneous use of an equivalent term Italia/Italic/Italian referring to people and geography, pre-Crucifixion, in the area now known as Italy at the least, or within any part of the Roman Empire. I'm satisfied it was so used, until someone comes up with evidence to the contrary.

Note — CE=AD