Your comcast remote is turning off the comcast digital box when you turn off your TV. You need to go into the comcast digital box settings if it has any and see if there is a stay on always option.
If not then you need to tell people at home to power off only the tv when turning it off, which should be accomplished with the comcast remove by switching to the TV mode on the remote and hitting the power button. If that still is turning off the box then you might need to get your self another remote like a logitech harmony, which can be setup to never turn off the comcast digital box when turning off everything else.
Is your slingbox connected directly to the back of the comcast box using component cables (red, green, blue?) or is it going through some A/V reciever?
Also by the sounds of it you need to make sure you have an IR sensor from the slingbox placed at the comcast digital box, if you are already have one there, make sure its lining up with the IR sensor on the box. If you are lined up then you simply need to re-setup your slingbox input and make sure you have specified the exact Comcast digital box you are using so the it can send the right commands for the remote control functionality..
Since you were using the internal tuner of the Slingbox Pro, you probably didn't setup the IR blasters. Now that you're connecting to a seperate device to view your television content, you'll need to setup the IR blasters with your set-top box. Once you setup your Slingbox with the IR Blasters pointing at the set-top box, and running through the video source configuration, you will be able control the set-top box using the virtual remote on the Slingplayer.