Windows users! I found the offending folder!
Navigate to C:\Program Files (x86)\Common Files\Apple and rename "Mobile Device Support" to "Mobile Device Support_old" (I just did this instead of deleting it in case it didn't work).
Next I went to Task Manager and killed "iTunes" and "iPod Service," unplugged my iPad, replugged it and launched iTunes. My iPad immediately showed up, and the iTunes Preferences window no longer takes five minutes to open. Hallelujah!
Edit: I don't seem to be able to add music to the device, even if I check "manually manage music on this device." So we're only part of the way there. We might have to find the exact file in that folder that's causing an issue.