what does “page” template really mean?

1) I’m templating my Blog and already templated most files. Just wonder what is the page.php for? Comparing code the index.php and page.pdp, the index.php do everything the page.php does (page.php only list the title, then “read more” link and number of page).

2) About the archive.php. Do I understand correctly: if I don’t have an archive.php, then the index.php will be used every time user query for posts in a month or catagory. If there is an archive.php, then archive.php will always be used, right?

3) I want to create a “master archive” page (a page list all catagories and number of post in each catagory so far, as well as list a list of archive by time, like this page: my old archive). Can I just create a new template and then just link to it?