Solution:
You can move your wordpress install into a new folder, no problem. Just make sure to update the paths in the wordpress config, otherwise the links won’t work correctly.
If you’ve linked to blog pages in blog posts, you’ll have to edit them manually.
See Moving WordPress and Moving a root install into it’s own directory (they’re different processes, it sounds like you want the first link, but read the second just in case).
If you want external links to continue to work, you could either (a) try to catch them and redirect with URL Rewrite rules in .htaccess, or (b) write a custom 404 error page in something like PHP that would redirect them.