Drupal 7: adding an image and a link to a user page
For images, just enable the Image module and add an Image field to your users just like you already did with other fields.

For a link, assuming you want that to actually display it as a link and not just as text, I really suggest you have a look at http://drupal.org/project/link. I know you said you don't want any contrib modules but again, there is no reason to not do that. Drupal only really gets it's power from contributed modules. Upgrading minor versions is pretty much always trivial and Drupal 8 is most likely still years away and Drupal 7 will be supported until Drupal 9 is released, so you don't have to worry about that for quite some time. Using link.module will certainly be better than trying to develop your own module. link.module has already been tested on 2000 other D7 sites (and 70k D6 sites..).

The Tab question is something completely different, so I suggest you create a new question for that. For a start, you will need to implement hook_menu() and add a 'user/%user/something' menu item of type MENU_LOCAL_TASK.