I Work out 3x a week, and am well aware of how important it is to give your body rest, but my question is, would push-ups on the 'off' days have any effect, good or bad, on the body?
I guess my thinking is that push-ups only use about 60% of body weight and if I do chest on mondays, would it be good to do push-ups just to stay pumped, or would that hamper the healling process?

not unless you are thinking of using it for some active recovery, and if you dont know what that is, then dont do them.

Stick to just doing them every other day.

You can weigh the load on your hands in the leaning rest. I weight 188 currently and the load is 121 on my hands, so that's about 65%.

If you want to add weight, load up a backpack.