Depends on hardware cloth. Is it 1/2 " x 1/2" 19 guage? If so, I tend to put them about 8-10 " apart - but allow for ones right on the corner...and adjust based on that. Larger (i.e. heavier - but smaller #, say 16 gauge, or 14 gauge)gauge can be slightly further apart. I do tend to be conservative...my brother-in-law watched me working one time when he happened to stop by on his way to somewhere else and said 'you are putting those too close - space them out more'. LOL fine - you build it for me, then! (he is a carpenter - but lives quite a ways away...no way he is going to build me anything!)Haven't gotten those yet anyway. I have to double check my hardware cloth, but I might have enough on hand if I 'seam' it, but won't do that until after I get the window built. How far apart should the scews/washers be?
Example: If your door is 30" wide, put one at each corner ( about 1" in from edge), then I would evenly space 2 more..so they will end up being roughly 9" apart.
For the pop door, it is smaller . but for the side (top to bottom), I would use at least 3 screws. one top left, one bottom left, one centered between. Ditto for the other side. If your door is bigger than what I am thinking - you can do more - but 3 would be the fewest # of screws I would use for affixing the plywood to the frame for that.
Note: It will also depend on just how tight you are able to pull the hardware cloth. I tend to use a heavier gauge, and am doing it all by myself, so sometimes have trouble pulling the 'waves' out of the wire from it being rolled (i.e. getting it to lay perfectly flat) - especially near the end of the roll/center, where it was rolled more tightly. I end up having to place the screws closer together then to keep the wire flat against the wood. (I would so love a fence puller and a good workbench to work on to pull things tight!!!!!)