I need to re-open a catalog that was created in Lightroom 5.7 but later opened in Lightroom 6, on the original computer with Lightroom 5.7. This is an institutional computer so I cannot upgrade Lightroom and they only upgrade once a year. How can I get my images with their current settings and my Lightroom presets back into this earlier version?
You can't open the catalog from Lightroom 6 in earlier versions of Lightroom.
Your Lightroom 5.7 catalog should still be in the same locaiton on your hard disk, you can open that in Lightroom 5.7, but it won't have any work done in Lightroom 6.
Or you can, in Lightroom 6, write the metadata to xmp, and then import the photo(s) into Lightroom 5.7, and then the edits and other metadata from Lightroom 6 will show up in Lightroom 5.7 ... except that if you used any new features in Lightroom 6 that are not in Lightroom 5.7, those will not appear.