I accidentally rejected and deleted from hard drive a photo I had modified. I got the image back from the recycle bin, but I had removed it from the collection as well and the settings are gone. Can I restore it from the catalog backups?
You can restore the image edits from the catalog backups. Once you open a backup, locate the missing image in it as explained here Adobe Lightroom - Find moved or missing files and folders
Then, use the "Import from another catalog"option in the file menu and import it in your new catalog.
Let us know if that helps.