Concerning the template, the description that you give me let suppose that K2 save the template in their /components/com_k2 directory somewhere.
As the /components directory is shared between all website, this is normal that you share the K2 templates.
So when you change something somewhere, as the files and folders are shared, this is automatically reflected in the other website.
Concerning the blank screen that you have when you try to install a template.
With which version of Joomla do you have this issue ?
If this is under joomla 2.5.x, retry with the JMS 1.2.77.
We fixed a problem related to the install with joomla 2.5 only.
See FAQ for the procedure to get the latest version
www.jms2win.com/en/faq/faq-joomla-multi-sites#cat-122