Hi Psaad,
Thanks for using WATS!
First, please let me explain why the user needs to have a minimum level of contributor in order to be able to submit tickets. This is because the 'edit_posts' capability is required to be able to submit a ticket. Indeed, a ticket is nothing else than a post with a post_type of ticket. So that's why you won't see the menuitem with subscribers because they don't have the edit_posts capability.
Now if you don't want your users to submit and modify posts, you could use the shared guest account feature. This is an account that you create and associate with the plugin through the options. This account will only be able to submit tickets, nothing else in the admin.
You could also use admin customization plugins such as the one you mentionned.
With regards to the ticket assignment feature, you have to set the option to indicate who can assign tickets. Then based on that, users may have to be logged in order to be able to assign tickets. The assignment can be performed on the frontend or the admin (if the user has the capability to submit/edit tickets).