If I set a date to be Monday the 1st of September 2014 at 12:59 PM, will this actually be handled as tuesday very early morning?
No, this would be treated as Monday afternoon, just before 1pm.
Early morning on Tuesday would be Tuesday 12:59am in 12-hour format or 00:59 in 24-hour format. And while it's true that people might also describe that as "Monday night", in my opinion, you have to stick to describing it as Tuesday unless you write it in speech-like form and use the word "night".
- The crossing point from Monday to Tuesday is midnight (00:00)
- The crossing point from AM to PM is noon (12:00).
- Midnight is the first instant of "AM" (edit: SOME places, not all), and can be written 00:00 in 24-hour format or 12am in 12-hour format.
- Noon is the first instant of "PM" and can be written 12:00 in 24-hour format or 12pm in 12-hour format.
When specifying a time for an event in Google Calendar, the time inputs are in 24-hour format so it's fairly unambiguous what time you are entering. If you enter a 12-hour format, such as "1:00", it corrects on blur to '01:00". Likewise "1pm" is corrected to "13:00".
Apple Calendar behaves the same way: