Quickstart and hosted service provider

I have a question. My hosted service provider provides a Joomla Install Script that works great. Can I use this and then use the Quickstart in order to gain the full effect of the Gavick Pro Template or do I have to use the Joomla install from Gavick Pro as well?

You are comparing apples and apples. You have to either use your hosts installer and install Joomla, then install the template and all the modules. Or us the Gavick installer to install the demo. You cannot install the demo on top of an already functional Joomla install.

Probably your service provider script install clean Joomla, the quickstart contains Joomla, modules and configuration so it is advised to install it in standard way, probably it takes less than 5 minutes.