Thanks for the help, everyone.
I tried the suggestion of changing the driver mode, and that did indeed cause Sonar to see the USB I/O. The USB connection caused a persistent whine, however, which is something I noticed when messing with it in the Windows Control Panel as well. I don't know what that's about. Thanks also for the info about ASIO, I don't know anything about these driver modes. I switched to WDM based on recommendations here.
So playing back an existing project using USB out, lots of stuttering (and that persistent whine). Also in a recording test, there were lots of pops and dropouts (which were reflected in the waveform, so they were during input). I unplugged the USB cable, the whine disappeared. I switched my I/O to the regular M-Audio (which is in fact Delta Audiophile), and left the mode on WDM. All seems to be happy now. Plus, my input drivers now list as "Delta AP 1/2) and "Delta AP Monitor" etc, where it used to be something else longer. My foggy guess is that switching to WDM is making the card use different drivers which have different names, yeah?
While I'm learning here, what's the difference between the 1/2, monitor, multi 1/2, 3/4, 5/6 and SPDIF? As mentioned, my card only has a single RCA stereo in and a single stereo out, so I don't understand why there would be 3/4 and 5/6. Maybe those just come because the drivers have them, but my actual card doesn't support them? So then what's Monitor and SPDIF?