You can do both.
If you create another mailbox user and promote it to primary system admin you will then be able to rename/delete the original one as far as I am aware. Guessing you cannot do it at the moment as there isn't another admin at the same level.
Alternatively, if you want to delete the incoming emails you can go to that account > Filtering > Content Filtering > New > Select either specific To address or From address then Next, complete the required info and the Next and choose delete...save and you are done.