Only spiders go for your chickens. If your water source is in a corner or on an edge of the coup then that can be the problem. Chickens hang out in there and spiders strike at them through the wall. Having the water in the center will solve that, but will cost you 100 xp per bucket to place. (CTRL click to place water in case you didn't know that one).
This is not exactly correct.
During Blood Moon, zombies will aggro everything, including chickens. Zombies with picaxes WILL dig to your animals.
Also, your walls are made up of cobblestone walls, not 100% full cubic meter solid blocks. Sometimes, mobs can "glitch attack" through corners or "less-than-full-block" blocks.
I suggest you make a second set of walls, about 8 blocks away and out from the first set of walls, to act as a "buffer zone". It might be kind of hard, what with your pen squeezed in between a tiny spot between extreme hills.
One other thing which might be happening to your chickens: they might just be drowning. Try to give them only still water instead of flowing water, especially "downward falling" flowing water is bad for their health. Top create a still water "spring water" source, CTRL + RIGHT CLICK with a water bucket over a 1x1x1 empty air hole. That costs 100 XP per source. A 2x2 water area is enough for a chicken coop of that size. You just do the diagonal,. then dig out the remaining corners of the 2x2, so... that is 200 XPs. Which is a good XP "investment" instead of dealing with all the hassles of flowing water.