Is it against HIPAA to allow patients' test and lab results to be exposed on a desk where other patients sit and can see name and test results of other patients?

Yes, it is illegal. The customers should not ever be able to see the name or any medical information about anyone other than themselves. The desk should allow for such privacy. If it does not, then there should be a workspace that is private where such papers can be worked with.

You are correct. Areas in an office that are easily viewed by the general public should have papers covered and computer monitors should be equipped with privacy screens.