Does anyone have any tips for renaming sidecar and .jpg files?
Yes. Absolutely.
Don't do it!MC names Sidecar and Cover Art jpg files. Leave them alone. The RM&CF function renames them correctly when you move the Movies, TV Shows, Audio Tracks etc. If it doesn't, something is wrong, or you are doing something wrong. Fix the issue, and then MC will look after the files.
If Cover Art images don't link to movies correctly, fix the Cover Art, don't re-copy the files from the source. Just right-click on the movie > Cover Art > Add from file, and navigate to and select the correct file. MC will take care of moving and renaming the Cover Art image file. Or you can use "Quick find in file / Cover Art directory, which works well if the file is named correctly and in the correct location, but just not linked to the movie record. You can do those processes in bulk, for any movies not set up correctly.
Unless you are fixing something that you screwed up by manually copying and renaming stuff, and you know what you are doing, don't go renaming Sidecar files either. Just rerun the "Get Movie & TV Info" process again. If you lose some tags in those files when you do that, and if that concerns you, just run the "Update tags (from library)" function for those movies / TV Shows.
One reason that Sidecar and Cover Art jpg files might not get moved is that you have imported them into MC. You do not want to do that. Check that your Auto Import settings for Movies, TV Shows etc. does not include importing images and xml files. Incorrectly set up Auto Import settings explains a lot of these sorts of problems.
BTW, the RM&CF function isn't really the best tools for creating a backup of your files. Sure, it can do it, but if you get the directory and/or file expressions wrong, MC is going to not just move the files between disks, but also rename the sub-directories they are in and the files. Plus if you copy the files, I think MC is going to import the copies into its Library as well, and you will have two copies listed. I haven't checked that. But anyway, a backup application or a file sync tool would be better for what you are doing.
Someone else has reported that the RM&CF function would only Move/Copy a few Albums worth of files at a time. Search the forum for that thread. There may be answers in it. The MC RM&CF function can copy thousands of files at a time, although that isn't really recommended. Break any large move operation down into smaller lots.
Also, make sure you are running the latest version of MC. There have been some fixes and improvements in the above features recently.
PS: I just noticed that you are running the Mac version, I run the Windows version, but in this functionality, they should be identical. But no guarantees.