The only real way to do such a thing is to put the catalog file and photos on an external HD and move the external HD from computer to computer.
1. Am I correct that writing metadata to XMP sidecar files will ensure that any edits made to a RAW file on, say, the Mac will be reflected on the PC, and that this happen automatically (i.e. LR is smart enough to look for the .xmp when viewing a photo)?
No, you are not correct.
You would have to manually instruct LR to read the metadata from the xmp sidecar files, Lightroom will not do this automatically. Forget once, and your Lightrooms are out of sync. Furthermore, some information cannot be stored in XMP, such as your edit history, collections, pick flags, virtual copies, and a few other things.
2. Assuming I do not have LR-PC and LR-Mac open at the same time (and am sure that DropBox has finished syncing), am I correct that the risk of file corruption will effectively be nil (at least when it comes to corruption caused by Lightroom)?
The risk of file corruption is NEVER nil no matter what system you use. However, it is for most situations, very very small. Again, this assumes you don't make a mistake somewhere along the way (and similarly Dropbox does its job without problem).
The only problem using both a Mac and a Windows system is that they address hard drives and folders differently.
Windows uses DriveLetters and the Back Slash ( \) and OS X uses DriveNames and the forward slash ( / ). LR looks for the Catalog file based on the PATH to it. On Windows it would be DriveLetter:\FolderName\FolderName\CatalogName. OS X would be DriveName/FolderName/FolderName/CatalogName.
Then you have how the external is formatted, if you plan on using an external drive.