Should I use one of my templates (which I would need to modify), or let ecart make the pages?
It's up to you, you can use a template, or not. if you select to use a template, you will be prompted to chose the editable region that the eCart items will be placed in.
Do I need to change any of the site or checkout fields, or can I leave them at their defaults? I looked at the help sections for those options, but I am still unsure. And will those fields need to be changed when the site goes live on godaddy?
the only one that would need to be changed is the Site Address, this should be changed to the address for your site on the live go daddy server.
Do I have to be live on godaddy to test transactions and such?
it is best to do your checkout testing on the live server.
One more question: What do I need to do to do to get a text field on every page displaying how many items in cart?
add a cart display manager to your pages, from the display type list, select Quick Cart.