If the files are all on internal disks*, the Fix Broken Links feature of Auto-Import will automatically do all of this for you (except remove the items from the Removed Items Database). It will just happen in the background, but if you want to force it...
Run Auto-Import Now?
Also, to be clear: Removing items from the Removed Items Database is only needed when you want to re-import them from scratch (generally for testing). Normally, this is not necessary, and MC uses this information to make the second import much quicker. It also allows MC to preserve metadata not available in the file-tags. If you're regularly moving things around and re-importing them, it might be helpful to uncheck the Ignore Files Previously Removed option as well (that feature is smarter than you might think, and doesn't just look at the [Filename]. I believe it will recognize the same file that has been removed and then moved around.
If you want to, for some reason, move files to a location not monitored by Auto-Import, then re-importing the files is still simple. While you still have Windows Explorer open after you moved the files, simply drag-drop the folder or files onto MC's Title-Bar area (where it says "JRiver Media Center 18" when playback is stopped, or the track title when it is playing). It will pop up a choice to Import the Files. Pick it, done. There is also the Shell Extensions in Windows Explorer, if you have those turned on.
Generally, I think you're overthinking it. Just move the stuff you want to move, and let Auto-Import clean up the mess.
If you have specific examples of where Auto-Import isn't cleaning up the mess, please post details so that it can be fixed. I'm certainly not seeing it (in fact, I wish there was some way to tune the Fix Broken Links feature to be a little
less aggressive).
* Fix Broken Links does NOT apply to network drives and external drives (USB, Firewire, etc) by default. It also looks at the root of the drive to try to determine if the actual drive itself is gone (in which case it does not remove the missing files), or if the drive is the same and therefore the files have been really moved or deleted. You can make it "just do it" (without these caveats) by changing the Remove Broken Links setting from Yes (Protect files on missing drives) to just plain-vanilla Yes. Be wary in this mode, though. If a file is gone (drive disconnects) for any reason, they get zapped. Also, note that eSATA drives are considered internal disks because eSATA doesn't provide a mechanism to "tell" that they're external rather than internal. All SATA disks look the same to Windows.