Some manufacturers weld on the expanded steel mesh to the bottom of screen doors. So, we bought a piece and put it on the inside with screws and found a plastic edging to keep the edge from being sharp or catchy. The window screening material remains in place attached with splines on the outside. The pet screen material is only as good as the splines and they don't hold against an animal pulling at it with their nails or pushing really hard. Hardware cloth wire/hail screen/rabbit wire would work well on the upper 1/2 because it would not obstruct your view.
For the window situation from the cat leaning on it...my only guess is the pushing may be too hard to do much about it or the spline is old and dried out. Maybe if you replaced it with new it might help. My only other choice would be to apply hardware cloth wire over the screen on the inside so it might ease the stress on the screen.
My main issue with screens is they slide too easily in the track and have no latch to hold them from being slid aside. I solved that my using blue painters tape at the window edge, catching in the screen frame but easy enough to remove and it does not leave residue.