The further apart the hands are, the shorter the distance of the lift. Hand width placement also affects sticking point. A closer grip makes it easier to move the bar off the chest but makes lockout harder. The reverse is true for a wider grip.
One of the main factors for if the bench press is more or less chest dominant is the direction of the elbows. Pointing the elbows straight out will stress the chest more. A tucked grip will result in more stress on other areas, the shoulders in particular. A good starting point is 45 degrees.
Touching the bar high on the chest will require more outward pointing elbows, thus stressing the chest. Touching the bar very low will mean tucking the elbows significantly, stressing the shoulders more. Each lifter need to find where they are the strongest but the basic starting point is the lower chest.
A good way to judge grip width and elbow direction is to look for "stacking". The bar should be directly above the forearm which should go straight down to the elbow (not in an angle). If the bar is behind the elbows it places significantly more stress on the triceps.
A high arch leads to less range of motion and thus the possibility to handle heavier weights. Arching should occur primarily in the upper back. Actively trying to arch the lower back is not as helpful and can be unsafe. For competitive powerlifters it's useful to try and gain flexibility to arch more. For people who just want to bench without competing, arching will still occur but it happens due to tucking the shoulder blades and tightening the upper back.
The legs should work actively during the bench press. Keep them tight from the start and push hard right at the moment you press from the chest. This is the most common way.
An advanced technique is to let the bar sink into the chest while in the bottom and "push press" it up using significant leg drive. This method requires a leg position that's not too far back.
The back is highly important in the bench press even after unracking and pulling it down. Once on the chest, keep the lats tight and try to create a “shelf” to push from.