I am very sure you will lose all profiles if you reinstall the Person App. However, if you backup the database for Person App and reload after reinstallation it MAY work if done correctly, you need to ask UNA Team regarding this.

Meanwhile, all that controls what you see and don't see are under Permission n Navigation. Be sure you get familiar with them first.

Also note that you see Edit for other accounts when you are logged in as Administrator or Moderator, this doesn't mean standard user will equally see other's Edit link.