Let me try to answer some of your questions!
Hiding the current post
Good point, never considered it. You could pass the post id manually as an excluded post, put if you want this to happen on many different posts it's not very elegant (or convenient). Maybe I should add another argument to the shortcode, like
hide_current_post. I'll put it on the roadmap!
Adding old posts
You could set sorting to random, that way, old posts will be included. This gives you very little control however. It really depends on what your needs are exactly for solving this problem. You could make a category for old posts that you want to push, and use that category, for example.
Tiles on pages
In principe, tiles work the same on posts and pages, so you should be able to add
[wp-tiles] to a page. Does this not work or do I misunderstand your question?
On category pages
Adding tiles to category pages is slightly more advanced, but is not hard. It does mean editing your theme - so if you are expecting updates for your theme and are not comfortable making child themes, this is not advised. Check out Child Themes in the WP Codex to see how to do it.
Once you have a child theme, the basic principle is this. You open the folder that contains your theme (you can use your cpanel file manager to navigate to wp-content/themes/THEMENAME/ on your server). Then edit the file called
category.php (or you copy it from your parent theme to your child theme) and find the code
while ( have_posts() ) : the_post();. Delete that line and everything up until
endwhile; (also delete that last line!).
Replace it by:
if ( function_exists ( 'the_loop_wp_tiles' ) ) the_loop_wp_tiles();
Also, see the FAQ of WP Tiles for more info.