No problem! The same course can certainly be in multiple categories. More on that here.
For the second part of your question - it sounds like you're looking for some customizations to your site landing page. Do you have any examples of what you're looking for that you can send us? If so, please send them over to our support team and we'll be happy to point you in the right direction :)
Is there a way for the students to view all of their courses by categories? I recently created categories but I believe that the categories are only visible in the collections page. I would like the student to be able to view the courses by categories once they are logged in.
This would require some customization to your site's HTML and CSS in order to display all your courses base on categories. You can definitely reach out to one of our experts here if you need a developer to help you out: http://experts.thinkific.com/collections/web-designers