to do that it usually does not use the lowest V2/Vr combination but rather slightly higher to much higher speeds
V2 overspeed (improved performance in AB lingo) takeoffs will improve weight compared to min V2 schedules until whatever limitation occurs to preclude further V2 increase.
Difference between Vr and V2 is usually only 2 to 3 kts
Which is why we don't normally concern ourselves with the V2/VR ratio - rather V1/VR
not enough even for the OEI case to just reach V2 at screen height
Can't speak to the NG but, for the Classic, rotating at VR, OEI, will approximate V2 at screen. AEO, of course, screen speed will be substantially greater than V2.
So in all practical application V2 and Vr are not fixed values
Absolutely correct save for the following caveat. VR is constrained by V2. At very low weights, V2 WILL be constant with changing weight (Vmc limited) until the weight increases to the point where V2 becomes stall limited. Overspeed schedules are used to exploit the improved climb performance at speeds modestly in excess of V2 min.