Repeatedly requests to change permissions, not syncing in user folder located on second drive

So the problem apparently turned out to be "symbolic links" (symlinks) hidden in a few minor apps I had floating in Dropbox. Lusil's reccommended actions were as follows:

"Symbolic links, Junction Points, or "symlinks", occur when folders are added to Dropbox that reference directories outside of the Dropbox folder or inside of the Dropbox folder but in a different folder. Because these symlinks may reference locations where Dropbox may have limited accessibility, they can cause various issues, such as high CPU usage, poor syncing performance, and permissions issues.

The problem you described may be related to such symlinks in your Dropbox folder, so could you please run the following command to determine if there are any in your account?

1) Open your Terminal app (Located at /Applications/Utilities/Terminal). You can also search for "Terminal" using the magnifying glass at the top of your screen
2) Copy and paste the following line into the Terminal: find ~/Dropbox -type l -exec ls -lah {} \;

This should give you a list of symlinks in your Dropbox folder and shows you the full file path to every symlink in your account.

If you do find symlinks in your account, we recommend that you remove them. You can do that by following the file paths that are provided when you run the command, and remove the files in your Dropbox folder by dragging them elsewhere on your computer (if you don't want them deleted). 

After you've removed all of the symlinks you can identify from the Terminal command, please quit Terminal, open it again, and run the command again to ensure that no symlinks remain in your Dropbox folder.

If the symlink pathway is part of a single file that is visible in your Dropbox folder on your computer, this symlink may be part of a container file. Container files look like regular files if you look at them on your computer, but they're actually wrappers that contain other file types inside of them. These may include files necessary for the application to run properly and load all the components that the file requires, with some examples of these types of files being iPhoto libraries, Aperture files or iWork files. 

In these cases, you may want to remove the entire container file, rather than just the symlink itself, as removing just a component of the container file could damage the container file and therefore you may have problems when opening it.

In general, we recommend users move the actual folders into the Dropbox folder and then symlink back to the original locations. Please just keep in mind that we don't generally support the use of symlinks due to the issues that can arise when they are added to Dropbox.

After you remove the symlinks, it can take some time for syncing to get back to normal. I recommend restarting your computer and then checking on the status of Dropbox. Please give the application some time to re-index and begin the sync process again."

...And, it worked. I am now syncing successfully. The Terminal command revealed a number of these symlinks embedded in a few minor apps (a couple of retro games and some console emulators), which I was more than happy to move outside of Dropbox. After a computer restart, Dropbox began syncing normally. Thanks so much for your help, Lusil!