If I don't see the "Full HTML" and "Filtered HTML" input formats, is it OK to add them manually?
By default, Drupal 7 includes plain text, filtered HTML, and full HTML text formats.

If you are not seeing these as choices, you are probably not logged in as an administrator and you do not have the appropriate permissions.

Non-administrators are not given access to full HTML by default because this can be a security risk.

Try logging in as an administrator and going to the permissions management page.  Verify the permissions for Use the Filtered HTML text format etc. under Filters.

Of course, if you need an input filter with a custom configuration, you can certainly create your own.  Just be careful to not give your users any input options they shouldn't have.