For where $(a,b)\preceq (c,d): \Leftrightarrow a≤c$ and $b≤d$.
As far as I'm concerned there are none of maxima, minima, largest or smallest elements at all.
Since there can always be a smaller $a, b$ or larger $c, d$.
But I'm not sure if I've understood it correctly. How can I prove that these (don't) exist?
You are correct that these do not exist. To prove it, just show that if I give you an element $(x,y)$ you can find a smaller and larger one. Then any element I claim is largest or smallest cannot be. When you ask for maxima and minima are you thinking in some local sense? Similarly, given an $(x,y)$ that is claimed to be a local maximum, can you find an element $(x',y')$ that is very close (in the Euclidean distance) to $(x,y)$ and greater?