How to add a user rank in their profile?
Use a combination of the Rules, Userpoints and User badges module, as further detailed below ...

Add a new badge

Have a look at this Rules example (in Rules export format):

{ "rules_grant_a_new_badge" : {
    "LABEL" : "Grant a new badge",
    "PLUGIN" : "reaction rule",
    "OWNER" : "rules",
    "REQUIRES" : [ "userpoints_rules", "rules", "user_badges", "rules_conditional" ],
    "ON" : { "userpoints_event_points_awarded_after" : [] },
    "DO" : [
      { "userpoints_rules_get_current_points" : {
          "USING" : { "user" : [ "site:current-user" ], "tid" : "all" },
          "PROVIDE" : { "loaded_points" : { "total_points" : "Number of points in all categories together" } }
        }
      },
      { "CONDITIONAL" : [
          {
            "IF" : { "NOT data_is" : { "data" : [ "total-points" ], "op" : "\u003C", "value" : "7" } },
            "DO" : [
              { "add_badge_by_name" : { "badge" : "1", "user" : [ "userpoints-transaction:user" ] } }
            ]
          }
        ]
      }
    ]
  }
}


It does retrieve, as the very first Rules Action (not Rules Condition!) the current amount of user points of a user. If the amount is at least 7, it will grant the user badge.

In this example I'm using the Conditional Rules module, though you could also rework the Rules Action to performing Rules Components (in which you add the appropriate Rules Conditions), so that you don't need this extra module (for more details on that, refer to the answer to "What&#39;s the alternative to using the Conditional Rules module?"). However, by using the Conditional Rules module, you can consolidate the logic for various user badges in a single rule, e.g bronze for 7 points, silver for 77 points, gold for 777 points.

The clue to make this rule work is the very first Rules Action "Load user points of a user". That's the alternative (work around?) for something like a missing Rules Condition to "check the current user points for a user".

Remove a new badge

If you want "a user's old badge to be replaced by a new one", then you should also add a 2nd Rules Action to the rule shown above. The action to be used is remove_badge_by_name (instead of add_badge_by_name).

Only grant a new badge once

To avoid "displaying a message each time when user getting more than 7 user points" (as in your question), just add a Rules Condition like "NOT User has badge". This is one of the Rules Conditions provided by the User badges module.

Display the badge

To actually "display the badge with the username at the top", you could use what's written in the Community Documentation about User Badges, i.e:


  To display user badges in your theme use:

&lt;?php
  if (module_exists('user_badges')) {
    print user_badges_for_uid($uid);
  }
?&gt;

  
  Note: $uid means 'a user id' here, not the literal use of $uid. Most people will use the uid of the node ($node->uid) or comment ($comment->uid) author.
  
  In Drupal (and theming), $node is a PHP object that represents the node, and $comment represents the comment. They are not always both available in every template file, so check the comments at the top of the file to see what is available. You may have access to other uids (user ids), and in these cases you can replace $uid with that value.