Hey,
For the 1st use case - You might have to set up a call forwarding from your phone to an exotel number. This will depend on your service provider, rates, etc. And there are some basic setup you can do for which calls need to be forwarded. There could be other 3rd party apps also which might help you with this.
2nd use case - You can add the “List Manager” applet as soon as the call is received. That way, regardless of whether the “Connect” applet is able to connect them with someone - they will first be added to your address book list.
3rd - There are some reports that you can generate. Exotel also sends out a daily summary email with some of these details. I am not sure if there is a live dashboard that shows who is on call right at the moment. but reports can be downloaded which will give you the details.
Hope this is helpful.