No, we don't usually re-date anythig, because that would create a terrible mess. Only in special cases the Proleptic Gregorian calendar is used, notably by Maya historians.

So the 10 days are legally missing, 4 October 1582 was followed by Friday, 15 October 1582. Thus Columbus made his landing in proleptic Gregorian calendar on October 3, 1492 (pre-1500 the difference was just 9 days instead of 10).

It does become interesting if you want to match things up to astronomical events. Someone describes that a solar or lunar eclipse occurred, but to match up, we need to use an astronomical/sidereal calendar.

There is a letter to Nature that fascinates me, it talks about one of the earliest recorded eclipses happening on 5th March 1223 BC. What does 5th March mean in real terms?

As you note, not everywhere changed at the same time. This also can become an issue (and is a major plot point in Umberto Eco's novel Focault's Pendulum).

In real terms, historians mostly use the original dates of the time in that particular country - as that is the best way to link records. It becomes an issue when different calendar systems were in use, as happened in Russia until 14th Feb 1918. You want to look at what happened during Napoleon's invasion or Russian participation in WW1, you had better have two calendars to track it when you look at the sources on each side.