Just set old router to AP Client mode if supported, and You’re good to go.

It depends really. I’ve install new WiFi bridge in 2016, and change UTP that goes by the wall in like 2020, as it change color and start to cracks. But other cable, which were installed in 2003 is fine, little bit stiffer than new one, but no cracks and still working well, same side of the house, may have little more shadow, but 20 years and it is regular cat5e cable, not outdoor one :P guess materials were better back in the days.

I agree with just painting those cables. Or to put those in cable conducts, not prettiest solution, but cables hanging loose is neither.