You will not be able to easily mix and match multiple dashboard tabs the way you describe it.
With a little leverage of the scripts available, you can probably set it up such that the Gadget Library only shows Gadgets of a certain type. But you would need to trigger off some session parameter which itself is triggered off which Dashboard you are looking at.
Almost impossible I think.
What can be done reasonably, is to have two dashboards, maybe a mashup with initially two tabs or a menu that switches between Thing vs. Template and on each of those you have a dashboard with its own library associated with it.