How do I override the default page.tpl.php? I have one page in my D7 website with significant deviation from the rest of my pages. In this page I will only display a limited number of regions and in a different layout.

What is the best way to override the default page.tpl.php? Can I create something like "mycustom-page.tpl.php" that gets called ONLY on a specific url within my site?
Depending for what exact content you want to make a special template there are diffrent solutions for that. Look at Drupal 7 Template Suggestions. Example for a single node: page--node--1.tpl.php.

In case the core template suggestions do not offer the flexibility you need, you can add more suggestions based on your own criteria. This is done in a preprocess function in your theme's template.php. For more information, see Working with template suggestions.