It is correct and formal to include "of" when writing dates?

Using 'of' is the standard form in Britain and those countries who speak the Queen's English. In spoken form 'the sixteenth of July' is used as frequently as 'July the sixteenth'. What we don't say in Britain is 'July sixteen'.

When writing we do not include 'of' nor 'the'. But a lot of people, me included, place 'th', 'st', 'nd' or 'rd' after the digits. e.g. 16th July, 22nd August, 1st March, 23rd April, 21st November etc.

Britain conforms to writing the short form of the date using the European method, which is used throughout almost the entire world except in the United States, i.e. day/month/year. so that 3/5/13 means 3rd May 2013, not 5th March 2013. I have heard it said that the US military follows the standard international method. If that be the case I can only think that it must lead to considerable confusion.

There are a variety of different date forms used whether in full or contraction.

This is certainly one of them. It's very common in Britain and Ireland, and rarer in the US, though it will be found there.

This matches the common short hand conventions, "the 1^st of June" matching the "01/06" or "1/6" most common in Britain and Ireland, and "June 1^st" matching the "06/01" or "6/1" most common in the US.

Note that in the US, it is common to speak of their Independence day as being "the 4^th of July" or the spelt out as "the fourth of July" or even capitalised to make it a title, "the Fourth of July". This is a legacy of that form being once the more common form in the US, and leaving its mark on the name of the holiday, despite the fact that the "Fourth of July" comes between "July third" and "July fifth"!