Requests in the Time Left Queue would be shown only when you had set Due Date/Time either through Workflow Assignment Values or SLA. To setup Due Date Time based on a specify criteria, Goto Setup > Workflow Settings > Workflow rule > Create rule with ...
Yes, you can split a customer's reply as a new request. Say, you respond to a customer's request and resolved his issue1. He then thanks you and emails you about another issue2. In this case, you can split this issue2 as a new request and assign it ...