Panasonic DVD/HTS connects directly to one HDMI input of the projector. This is for DVD playback.
Fire box HDMI out connects to an HDMI input selector with audio extraction. The selector HDMI out goes the other HDMI input on the projector. Any other sources you want to watch also connect to the input selector with HDMI. The optical audio output of the input selector connects to the optical input of the DVD/HTS.
If you get an HDMI input selector whose IR codes are already in the Logitech database that will make it easier to program. It need to have discrete codes for each input rather than cycle through the inputs.
If it makes programming the Logitech easier you could connect the DVD/HTS to the input selector as well and only use one...