Inside the front cover of my passport, the following statement appears;
Her Britannic Majesty's Secretary of State requests and requires, in the Name of Her Majesty, all those whom it may concern, to allow the bearer to pass freely without let or hindrance, and to afford the bearer such assistance and protection as may be necessary.
I would think that the Queen herself doesn't need her Secretary of State (which we know commonly as the Foreign Secretary) to ask on her behalf that she be let into a country.
All a passport is in essence, is a document from the government saying you are who you are, and you have some authority to leave the country. The UK Government is de jure the government that represents the monarch - the executive arm, if you like, of the sovereign.
In common law, she doesn't require any of this documentation because of the royal prerogative. This embodies the power of the monarch, and in the United Kingdom, the royal prerogative can only be changed using a special procedure.
To just take each item of your list in order though;
There is no such thing since the scheme was scrapped in 2010.
There has never been such a thing as an EU passport, but she hasn't got a UK one either.
- A diplomatic passport of a sort
- Any ID card replacement that may be used for personality identification, such as a driving licence.
No, and she doesn't need a licence to drive a car either, for the same reasons that she doesn't require a passport. She would effectively be issuing it to herself.
- Any royal certificate or diploma
What she does have, however, is a birth certificate. When she was born she obviously wasn't the queen (indeed she wasn't expected to become queen in the future as her uncle was the heir to the throne), so the government - through the local authority - issued a birth certificate. Monarchs also have death certificates, as when they die they cease to be the monarch.