I was asked to evaluate help desk solutions for the IT department because email isn’t a great way to track problem tickets.  The other needs being
1) the ability to track open issues, 2) a checklist system that provides real-time feedback to people 3)customizable 4) ease of use 5)self service portal. Is there any helpdesk software that fits in all these features?

There are a lot of helpdesks which are available on the net. I would suggest a web based service called pathworks software. They have solutions for your requirements. It has a very easy to use interface and has a guided self service portal. They also have complete email integration and interactive checklists for clients which give the clients a visual representation of their cases. We have been using it for the last two months and there has been no problem so far.