Push-up routine/tips?

Should I go for fast reps or slow ones when doing push-ups?
And how often should I be training if my goal is to max out my total count?

I do my pushups at a moderate pace—slow reps, if you want to call them that. I don't bother with them at the gym; I just save them for when I'm working out at home.
I’m not trying to crush myself with some intense, high-speed set just to pass out. Instead, I do three sets of 40, moving slowly to really feel the burn in the muscles afterward. I used to do fast reps, but I eventually realized that going slow is actually more effective. It's harder for me that way, and I get a much better mind-muscle connection.
Of course, I pair all this in with my other exercises.

Do you go for wide or narrow push-ups?

It’s pretty straightforward: if you go with narrow-grip pushups, you’re hitting your triceps, whereas wide-grip variations shift the focus right onto your chest.

Do pushups actually help strengthen my back?

No.
Push-ups build your chest, while back and chest exercises work in opposite directions. When you're doing back exercises, you're pulling the weight toward your torso, whereas chest movements require you to push the weight away from your chest.
Plus, push-ups hit your shoulders too.

1. Is it actually worth grinding out slow-motion pushups if I can only squeeze out 13 reps before hitting failure?
2. How much more focus goes into the chest when you elevate your feet on something like a bench or a chair, and what does that do to the rest of the muscles?