To disable date output or categories may involve editing the theme files directly. Each theme may handle this differently. Usually there is a line that looks like this:
<?php the_time('D, M j, Y'); ?>
<?php the_category(', ') ?>
You can comment those lines(which means in php they won’t output or do anything) like this:
<?php //the_time('D, M j, Y'); ?>
<?php //the_category(', ') ?>
Obviously, if you are not comfortable editing template files, I do not recommend digging in to those.
You can choose which posts can have comments at the bottom of the post creation page in the “Discussion” panel.
Your home page is probably index.php. This is a dynamic file that pulls the most recent posts into your home page. You would need to have a pretty good understanding of WordPress to want to edit that file.
You can set ANY page as your home page btw in Settings->Reading.