1. Yes, F20 will be more desirable as you are off the ground sooner so in case of gusty wind or wind shear, you will have more margin to accelerate and still clear obstacles.
2. Yes, continuing takeoff after engine out at speeds higher than MIN V1 will meet all obstacle clearances, rejects can be made up to MAX V1 (runway margin will reduce as speed increase toward MAX V1, after MAX V1, an overrun is likely). At reduced thrusts, Max and Min V1 will become closer together as reduced acceleration means a higher speed is needed at engine failure to continue (higher Min V1) and reduced acceleration means more runway is used to reach Max V1, so Max V1 has to be reduced to have enough runway to stop.
3. If the turn-around was quick, F20 departure gives lower V1 which means less energy into brakes, so less brake cooling time is needed if the brakes were very hot from the previous arrival or from taxiing. There will be less chance of the fuse plugs on tires melting as the larger runway margin can allow autobrake to be disengaged and manual braking done at a less aggressive rate if enough runway remaining/deceleration is good enough. Advantage of F15 is higher rate of climb which can be useful when encounter downdraft, especially when engine out (of course if downdraft is expected before takeoff, not change to F15 entering downdraft, unless captain's judgement says otherwise - with enough altitude, caution and good technique, switching to F15 and accelerating while sacrificing a bit of altitude can gain higher climb rate afterwards, this should not be done until initial obstacles are cleared as distance taken to accelerate will reduce obstacle clearance). A strong downdraft and engine out, F15 may allow level flight while F20 may cause slight sink. F15 may also allow a slightly better assumed temperature, causing less wear on engines, but safety considerations should come before this. There may be a recommendation on flap to use when windshear or downdraft are possible (of course don't takeoff into a windshear if radar detects it or it is reported) in the FCTM, I've seen it for some Boeing models.
4. MAX V1/VR means that MAX V1 is limited by VR speed as V1 cannot be higher than VR. If VR is high enough such as in F15 case, MAX V1 is limited by stop distance available (field length) (higher V1 won't allow a stop to be made at speeds close to V1). But in F20 case, V1 cannot be increase more than VR so it does not reach the field length limit speed. (example: speed after VR in F20 case, you still have enough margin to stop if you reject, but you will have already rotated and decide to go). Seeing MAX V1/VR vs FIELD LENGTH for another flap setting in the MAX V1 line means the MAX V1/VR flap setting gives more stop distance margin as the plane rotates before reaching the reject speed where all the runway is used to stop.
Hopefully this helps and feel free to correct me anywhere if I'm wrong for the experienced folks out there.