There are plenty of DVI-D to HDMI cables. The only problem is that DVI-D can't transfer sound, while HDMI can. Not sure whether this can cause problems when broadcasting or not.

As for broadcasting to Justin TV with AverTV HD DVR, it comes with AVer MediaCenter. It allows to to simultaneously enjoy HDMI or Component video in one window and in the other window watch video from composite video or S-video sources.