Yes.
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There are an infinite number - all of the same length.
There are infinite diameters in a circle all of the same lengths.
Yes but they are all the same size
-- Every circle has a diameter of some size. -- All of the diameters that you can draw in the same circle are the same size. -- The smaller the circle is, the smaller its diameter is. There's no minimum size. -- The larger the circle is, the larger its diameter is. There's no maximum size.
This question does not make sense. All chords are not, in fact, diameters. Actually, only chords that pass through the center of a circle are diameters.