Hi that 3rd photo is the same as the 2nd one, not sure if it helps much. I recommend you work out which cables go to where, plug the router into one of the patch ports in the cabinet and then take your laptop and check all the ones in the AV closet till you get a connection, rinse and repeat and record which are which. For a start disconnect all the yellow patch leads going to the patch panel.
Once you have established what goes where and presuming the AV location is good for your router (ie gives good wifi coverage) then place the router there, using one of the data links connect the ONT port1 to the WAN of your router. You can then use a 2nd data feed to take a LAN port back to the garage, then you can use a switch to feed other outlets that terminate in the garage cabinet.
My guess is not all the eight data outlets in the AV closet go back to the garage, instead they may go to other rooms, I presume there are other single and double outlets in other rooms? If so its possible some are fed from teh garage and some from the outlets in the AV closet, you will need to work it out. Also the HDMI leads in the AV cabinet, I presume they go to a wall mounted TV nearby, there also appears to be some cat5 data cables, I presume these also end up at that TV, in which case you can feed the ethernet port on the TV for better streaming.