To me, Combo and mobility are the two key components to get a high score. Time bonus may seem nice, but its impact on the final score is not as significant as the two others I mentioned. Getting the frog mask makes getting a big combo easier. I usually focus on getting the largest combo possible while refraining from too much use of guns, ignoring how many tries it takes me.
In killings, you simply get points for how you killed your enemy. You get the most points from kills if you are unarmed, and less using melee weapons. Fire-arms do not give a lot of points but they are nice to get a big combo. Exotic weaponry such as the drill gives the most.
Boldness should be how reckless you are. If you put yourself in dangerous situations where the enemies try to attack or shoot you, boldness goes up. I am not completely sure about this though. Use of firearms can have a direct negative impact on your boldness score.
Mobility drops if you are staying put all the time, waiting at doors and behind corners. Moving about from room to room without stopping ensures a high mobility score.
Flexibility is supposed to be how you make use of the weapons available to you. If you kill every person on a floor with fists only, you get a high killing score but low flexibility. Constantly changing weapons from what's available improves flexibility. If you are never getting any flexibility, you may need to patch your game.
When you manage to clear an entire floor in a single flowing combo, you are pretty much guaranteed the A+ rating.
Another way to get very high scores is to use non-standard weapons such as the drill which gives 5600 points per kill.
Edit: The points awarded from Drill-kills have been reduced to ~1000 points in the latest version.
I've never gotten anything other than 0. My guess is that they didn't finish adding the code that measured it, and it would have something to do with how much you used and reacted to the randomized spawns (weapons on floor, held by enemies). That sounds difficult to assess fairly, and usually the randomization doesn't affect your run that much, which, to me, explains why it's not working. I bet they'll just remove it and maybe add a different category later in a patch.
Again, total guesses of mine.
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It saves the chapter you're in, allowing you to go back and go through it from the beginning, so when you move to a new chapter you'll have that auto-saved.