I'm using Asana too and it's definitely a workable solution for the no cost. You can't set up an email forwarding system. Where staff sends you email with specific subject "Tech Request" and then an email management rule to auto forward from your email to email@example.com
which would then create a task for you.
For management reports... I'm not totally familiar but I thought Asana had some report options (or it might be on the new paid version) and what about the Dashboard view, could mgmt login to Asana and have a Dashboard that you set up for them to see progress on work? You may have to create some general Projects or something to put it on the Dashboard.