Is there a way to put different background colors to each query on a dashboard that have multiple queries side by side?
No, not easily. It’s possible with custom code, but you’ll have to dig around the bowels of infragistics to do it. Also, that becomes obsolete once you do to Web-UI, so not really value added.
Would there be an easy way to add some kind of border around each query then to separate all the data being shown?
I don’t know of a way to add borders. And I don’t know what you mean about “separate the data being shown”.