I'm running Kubuntu 15.04. On my desktop each icon is displayed in a gray rectangle along with its name, and the icons themselves are quite small. This is different behavior than what I saw in 14.10. How can I get rid of that gray background and have the icons displayed full-size? I suspect some issue involving themes, but I have no idea what it is.

It is definitely dependent on the plasma theme used, as it is in KDE4. Breeze happens to give the plasma widgets on the desktop that background you are seeing.

You can experiment by trying new themes, and see if there is one that fits the bill for you, or at least provides a transparent background for the desktop widgets. From System Settings, you can go to Workspace Theme >>Desktop Theme and view/add/select Plasma themes. In the Details tab, you can even set a specific element to a different theme. In this case, you would change the Theme Item Widget Background (or was it Translucent Background? ) to a different theme you have installed.

Note that this may not work unless you have Plasma 5.3 from the kubuntu backports ppa - I could not get this to work until I upgraded, on multiple machines.

As for size, the icon widgets should be able to be resized like any widget.