1) Can I just delete the Lightroom folder which contains the a Backups folder, and the following files: LR4 Catalog Previews.Irdata, LR 4 Catalog.Ircat, LR4 Catalog Ircat-journal, and LR4 Catalog.Ircat.lock?
2) How do I "sync" up LR4 on the new Mini to existing picture foldesr on the old hard drive, essentially making a new catalog?
1) yes you can delete those if you really do not ever wnat to use them again
2) you would have to import every folder/image and generate previews for all those images, all over again
Why do you want to do that, why not just copy the existing Lightroom folder to your new machine, you could do this via your external drive as a 2 stage process; copy from old machine to ext - copy ext to new machine. If you have a lot of images could save a lot of processing time compared to starting from scratch.