Is there a way to display different search results for public and registered users? I have 2 groups of users - those who want to share their information and those who only wish to disclose them to the registered users.

You can create 2 menu items of type Easy Profile -> User List with different Access (maybe you need to create Access Levels before). On these menu items you can set a static filters with parameter "SQL Custom Where", some example at: http://docs.easy-profile.com/article/sql-custom-where

For example you can set first menu item with only users from a usergroup with this syntax in SQL Custom Where parameter:

group_id IN (5)

and second menu item with users from another usergroup

group_id IN (9)

You can find usergroup ID from Administration->Users->Groups

those who want to share their information and those who only wish to disclose them to the registered users.

For each field there is a possibility to set "Visible for" parameter, so you can set your field to show value only for some users. For example if you set this parameter to "Special" only owner and Admin can see this field. For more info read the tip on this parameter.

Is there a way to input an information (e.g. date of birth) that ONLY a user him/herself would see but would NOT be able to change it?

You can set a field to not show on edit profile page