All of these changes may be made in your header.tpl file in templates/theTemplateYoureUsing/header.tpl
In the default template, the section of the header when not logged in is lines 120-149. Where exactly you'll want to add the order link is up to you.
To remove the affiliates link from the nav bar, you may simply remove the code that generates that link. Lines 100-102 (for when a customer is logged in) and 132-134 (for when not logged in)
To add new links to the navbar, you'll simply need to add additional points to the list. I'd recommend modifying the affiliate link you want to remove as your first link. That way, you'll have a good idea of what the formatting will need to be.
I recommend you check out our documentation regarding client area templates. All the templates are written in a template engine called Smarty. Their documentation is fantastic - easy to read and covers all the bases from simple formatting to logic application.
One thing to note is that editing the nav bar, especially if you add too much, can result in the site looking a little wonky. You may need to adjust your CSS as well to handle the newly-sized bar.