What's the historical origin that makes us write 1.000.000,00 and not 1,000,000.00 (like in English) or 1 000 000,00 (like in French)?
Belgium, Denmark, Germany, Italy, Romania, Sweden and most of Europe use the dot as a thousands separator, but the space is adopted as well. Even if it must be said that the International Standard recommends spaces in lieu of dots or commas.
As stated on this wikipedia page, before the invention of printing, a line ( ¯ ) was used over the digits. After printing, the countries/nations chose marks for convenience and according to what others had already chosen to avoid confusion. This means that it basically was an arbitrary decision, no particular reasons were behind it, apparently.
You shouldn't, except perhaps when writing by hand.
Separating thousands with periods is cause for ambiguities. When I was young, I was taught to use a raised dot like in
123˙456, not a period; probably different teachers use different conventions. The main reason I see for a dot instead of a thin space is that usually numbers were written on carta a quadretti, so one digit per square.
When printing, use a thin space:
1 000 000
It's just as readable and not confusing. I'd even say it is more readable.
This is recommended by UNI/ISO, as this quote from the book by Roberto Lesina, Il nuovo manuale di stile, Zanichelli 2009 (p. 174):
La spaziatura è il solo metodo universalmente corretto per la separazione fra migliaia. L'uso di altri separatori, quali il punto [...] può portare a grossolani malintesi. [...] Per questi motivi, anche l'ISO sconsiglia l'uso del punto e della virgola come separatori di migliaia», citing «ISO 31-0, Quantities and units, General principles.
I know of no “official” statement that baseline periods are to be used in Italian.
Now, why did this convention enter into common practice? The answer is really easy: typewriters didn't have raised dots. The influence of typewriters on the way we write now has been great: for tens of years, we had only printed, typewritten and handwritten texts, the middle category being prominent in business.
The raised dot had the same fate as ‘È’, that became
E', or the digits 0 and 1, that became
O (oh) and
l (ell), or the guillemets: disparition.
The question might be phrased: why should we use undirected
" instead of inverted commas or guillemets? The answer would be the same. By the way, guillemets were hardly found on typewriter keyboard and they are disappearing.