To answer your second question: No, it's not possible to access a stamp from
a server location. It has to be installed in the local Stamps folder.
As it has been said, you can't provide a "stamp server". However, you can provide a stamp _service_, which would do the stamping server-side. You can use libraries to create such a service, or go with commercial products.
You need to load the PDF as a dynamic stamp into your copy of Acrobat and then you need to edit that copy of the stamp and add the interactive script.
Creating a Custom Dynamic Stamp by Lori Defurio
Dynamic Stamp Secrets by Thom Parker
The Response Box: Part 2 of 5 on Popup Windows by Thom Parker