1) I have image files in folders of the form c:\users\name\pictures\yyyy-mm-dd and d:\photos\yyyy-mm-dd. I want to move all of the folders that are on c: to d:, so that all the images are under on d: in the appropriate d:\photos\yyyy-mm-dd folders.
LR is a decent tool for simple folder movement and copying, and it's generally a good idea to do such things from within LR, as it won't then get confused about folder structure. However, automating a task of this complexity is, I think, way beyond it.
Do I just go to LR4, right click on the pictures folder under c:, choose Update Folder Location and type c:\photos?
No. That would be to tell LR that the photos were in fact in c:\photos, which isn't true. "Update folder location" merely updates LR's idea of where the folder is: it doesn't move or copy anything.
Will there be a problem if some of the yyyy-mm-dd are in both c: and d: or some image files are duplicates?
Possibly; it depends what you use to do the copy.
One of the folders in c:\users\name\pictures is Lightroom, which contains the LR catalog, previews and backups. Will this move when I use LR to move c:\users\name\pictures? If so, how do I move it back to c:?
No. Using LR to copy folders won't affect the catalogue location.
I could go through the folders one by one, but that would be tedious.
I don't think you've any option. On a Mac (sorry), you could use Applescript or a shell script, either of which would be fairly simple and could ask you to address any problems with duplication. There may be something similar on Windows. One caveat: if you copy the files outside LR, you'll lose any edits you have applied to them within LR, unless you save them in sidecar files and copy those as well (I think: I'm open to correction on that one).
I suspect all you can do is to set aside some time and copy all the files manually within LR.
2) There are a number of folders in d: with no files. I can right click on each one and choose Remove, but is there a less tedious way to have LR4 remove all empty folders?
I don't think so, no. Again, you need a Windows equivalent of the Mac/Unix shell command "find", followed by telling LR to Synchronize folder.