There are various ways to search Facebook without logging in. The most complete function is using a search engine like "Social Search" or "Open Status Search," which allows you to search all social media activity that has been posted publicly from a specific user or brand.
You can install an add-on feature in the Chrome web browser that allows you to check your Facebook notifications without logging into Facebook. This feature will inform you of any notification that you received on your profile via email. The add-on is called "Facebook Notifications," which is easy to download on any web browser.