I am thinking about replacing my broken pump with a 1HP dual speed Hayward Super Pump. Does anyone know if it will be able to push (not pull) the water up 8' in low speed?
Yes...it is a closed loop not open to atmospheric pressure so you could put the pump 80' on the top or bottom of the loop...no difference.

Can you still have the filter in the filter mode on low speed? Will my in-line automatic puck chlorinator still work on low speed?
1. Yes 2. Slowly...although most of us here don't like in-line chlorinators


Also, will I have any trouble with the 1hp super pump being too big for (1) 1 1/2" dia. skimmer intake and (1) 1 1/2" dia. return?
Wow! That's not nearly enough pipe, skimmers, or returns for that size pool. I see why you're remodeling.

I'd probably get the same pump and live with it for a year, tho. The problem you may have is not being able to stay on "low" 24/7 because you won't turn your water as often as you should.