Repairing concrete floor in basement entryway

I've got a basement entry way that's got some serious pothole(s). The problem is that the water runs down the outside steps and puddles on the floor. This then freeze/thaws and creates the potholes. So I need to fill the potholes and add some slope to the floor so that the water runs down the drain properly. I gather some cement:sand at 1:4 with some PVA is the way to go? I've also read that I should paint the floor with PVA before I start? The deepest hole is a good half inch and I want to at least an inch to the outside walls for a 1/4" slope to drain the water.
Problem #2 is that the door to the house opens into the entryway, so sloping the floor along this wall will be a challenge. Of course it's the same wall that the outside door/stairs is on. I was thinking I could leave a pie shape to allow the door to open, but this might look like crap, present a tripping hazard and allow any water to chip up the new cement.
I was also thinking I could put a gutter (ABS cut in half) under the outside door (it's one step off the floor) and run a drain pipe around the outside of the room to the floor drain. This would not catch all the water (walls are damp), but I think it's 90% of the problem. I'd still have to patch the floor, but sloping could wait for next time, if ever.
So any thought on:
1) The best slurry for patching sloping 2) How slope under/away from the outswing door. 3) My gutter idea.
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