Kashi,
Unless dumbell overhead presses and benchpresses (maybe on the stability ball) with your 10# weights are really too easy for you, I would focus on those first as better means of building your pressing strength initially then the assisted push-ups. Push-ups are very hard if you are heavy, but get alot easier the stronger and lighter you become. I am still heavy enough that 20 push-ups is a near death experience, but I am sure that will be banging out sets of 50 when I get down to 240#. For me a pushup is about equal to a benchpress with 2/3 bodyweight, so a regulation push up is a very lofty goal at your current weight, although it should be realtively easy at your goal weight.
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2/ This is kinda gross, but whenever I do pushups, when I go towards the floor my back fat/skin is pushed together (I guess...?) by my shoulder blades and it hurts! Any way to change this, or should I just wait until I'm smaller to be doing pushups?
Sounds like a form issue caused by the pushup being too hard for you. Don't sag on the way down, but keep everything tight and you shouldn't have any pinching, if I understand the problem. Don't try to shorten the ROM by pushing your chest forward.
Robert