The best suggestion I could make is to put the router on a UPS which you have already done. The UPS will do more to protect from surges than any plain surge protector.
As for protection on the coax, absolutely not. The coax has no connection outside your house and does not need a protection device, or if your ONT is outside then the coax has a lighning protection device in series it. Many have reported that adding surge protection to the coax causes issues with guide and VOD. Same is true for the ethernet since the damage occurred with a surge to the router which the UPS should now handle.
I have had my router, PC, and DVR all on UPSs for years and haven't had any issues with damage due to surges, brown outs or any other power related problems.