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The default template comes with every installation and can not be modified. No matter what theme you select, it will always use the default template, unless otherwise specified through the admin panel.
Once you clone your default template, you can make changes to how certain elements on a page will be displayed. For instance, if you want to change how your redirect looks, you would modify this in the template section of your admin panel. If you have the default template selected in your admin preferences, and have modified a clone of your default, your modification will not show up until you have selected the cloned template from your admin panel, if you follow what I mean.
In general, theme selection will not affect your template selection or layout. You must specify which set of (cloned) templates you want to use for your site overall. On rare occasions, theme develeopers have included a default template set that enhances the overall theme output, but this is very rare. Of course, you do not have to use the (rarely) included template files that (may) come with your theme.
I hope this clears up any confusion you might have had over the difference of themes, templates and their usage. Please, feel free to post any further questions you have, as I'm sure I havn't quite touched on everything you were asking in my general overview.