Open theloop.php (using Presentation/Theme Editor) and look for:
<?php if (is_archive() or is_search()) { the_excerpt(); } else { the_content("Continue reading '" . the_title('', '', false) . "'"); } ?>
Change it to:
<?php if (is_search()) { the_excerpt(); } else { the_content("Continue reading '" . the_title('', '', false) . "'"); } ?>
If you want to change it to simply Read more >> then use:
<?php if (is_search()) { the_excerpt(); } else { the_content("Read more >>"); } ?>
If you do not want this ([…]) even in search page then use this:
<?php the_content("Continue reading '" . the_title('', '', false) . "'"); ?>
In categories and search view, by default, the excerpt is shown which doesn’t display the links. Follows the steps above (question 1) to solve this. It takes care of both the problems.