You open a number of considerations, in particular are you considering heavy aircraft, commuter, or light twins. The rules, both for design/certification and operation differ with the class of aircraft.
V1 = Vr must be nice (no 'dead man's zone'), but when the two speeds are not the same, I thought V1 was the speed at which you have enough runway to stop in but not enough speed to fly - however many engines are running?
The idea of these speeds is a bit conceptual but it's stood the reasonable test of time. Without too much of a pad available, below V1, with a failure, you should be able to stop. Similarly, above V1, you should be able to go. That's a bit idealistic, and there are a few ifs and buts in the wings but it more or less works pretty well most of the time. There's no guarantees with this stuff, only probabilities and if you push the presumed boundaries what fat there is can rapidly disappear and the folk can end up in a world of hurt.
The certification process tries to balance a bit of conservatism with a bit of commercial practicality with the aim (hope ?) that folks get it all together on the day. Hence the emphasis on over training so that the pilot doesn't have to waste time with cognitive effort directed toward the basic mechanics of doing the exercise on the day.
I was always under the impression that V2/positive rate of climb is the point at which you could execute a safe return even with an engine out?
V2 is geared to the situation where you do have a V1 failure. The need, then, is to make sure that you fly the aircraft in a manner that, with all engines going, you make a flightpath which is within the engine out flight path.
So, with a V1 failure, for the Flight Manual numbers, when you get to V2 you should be well-placed to recover the aircraft. The gear is addressed within the segmented takeoff sequence. If you didn't have a failure (the normal situation) it all becomes a bit of a doddle and much easier than with the engine failure.
But I'm not a pilot, only a reasonably well-informed enthusiast.
That's fine. We all try to be well-informed enthusiasts. Some of us fly aeroplanes as well.
I see my good colleague, GF, beat me to the punch. Just to clarify one of his observations. We don't set out to increase the actual speed if we have a failure. If the failure is at the Flight Manual V1 point, you should end up at very close to V2 when you get to 35 ft, providing you fly the aeroplane like the OEM says you should. Obviously, most of the time we don't have a failure and, if we do, it usually is not at V1. So, if we have a failure a bit after V1, we have the benefit of the all engine acceleration up to the failure point with the result that we end up at 35 ft with a speed a bit above V2. Post the O'Hare DC10 mishap, the idea is that, when we get to 35ft, with a failure, if we are
(a) below V2, then increase to, and maintain, V2
(b) at V2, maintain V2
(c) between V2 and, usually, V2+20, maintain the speed you have
(d) above V2+20, pitch up to reduce to V2+20
The reason we don't accelerate to increase the climb speed (and get a bit better climb rate) is that, while accelerating, we are likely to end up in the rocky bits which is not a good life options strategy.