Θέλω ένα κρύο καφέ without Εγώ should be considered as a right answer also.
Actually, it should be έναν, but your point is valid.
Thanks for noting this, D_.. I paused for a second because I thought έναν was correct for accusative but the end of the clause had nominative καφές so I went with ένα κρύο καφέ
I also agree! Who else could θέλω possibly mean other than 'I'?
I put ένας and not έναν. Why it has τo be έναν?
Because it's the direct object -- the thing that is wanted.
So it has to be in the accusative case: masculine accusative έναν, not masculine nominative ένας.
is a frappe not a cold coffee?