Attempting here to stay strictly within the framework of the original post, which asks - "Can anyone explain in simple terms the effect of changing V2 on the performance especially with regard to after takeoff obstacle clearance?"
If V2 cannot be less than 1.2Vs (or 1.1vmca) then surely it is a function of configuration and weight.
Yes it is a function of configuration and weight. Up to a certain weight (for a given configuration), V2 will be constant, being constrained by the 1.1 Vmc requirement. Above that weight, V2 is governed by the 'minimum of' 1.2 Vs rule, and will increase with increasing weight. It NEED NOT be equal to 1.2 Vs, there is no upper limit specified to the people doing the performance certification, although practically, it rarely is seen higher than ABOUT 1.35 Vs. 1.2 Vs is inevitably on the low side of Vmd, thus available gradient will be less than optimum, but a good trade-off against runway length. Obviously a V2 of 1.35 Vs will require more runway than 1.2 Vs. If increased runway lengths are available, then the operator may avail himself of the "Increased Speeds" V2, but only if AFM approved. (It is for this reason that operations using V2min are often permitted to carry out 2nd segment Climb at speeds such as V2+10 or V2+15 or so, provided that the aircraft has already reached that speed at the time of the failure).
"So, if we increase the weight (all other factors constant) we increase the V2 speed but what happens to our 1st, 2nd, 3rd obstacle clearance performance?"
Given that all other factors are indeed constant, the 1st segment gradient will be less (but not necessarily longer), the 2nd segment gradient will be less, and the 3rd segment will be longer. This is normal. If we are seeking the Maximum Permissible Takeoff Weight, we will steadily examine increasing weights, with their associated increasing speeds and declining performance, UNTIL the Takeoff is limited (either by runway length or minimum obstacle clearance). Under the same environmental conditions and configuration, but at a lesser weight than the limiting weight, performance will EXCEED the runway length and obstacle clearance requirements. In other words, you're better off. This process of steadily increasing the weight until limiting is normal procedure in establishing RTOWs.
"If we use more flap our stall speed reduces so our V2 would be less? What does this do to the climb gradients?"
Yes, V2 will be less (provided that you are above 1.1 Vmc), and the climb gradient will also be less. against this, the required Runway length will be less, and this may be the more limiting factor. In fact, one of several good reasons for using a greater Flap settings (there are others) are that provided obstacle clearance is not compromised by the reduced 1st, 2nd, and 3rd segment performance, improved runway performance is possible. In short, operation from a short runway may be possible, whereas it is not possible or not viable at the lower Flap setting.
Regards,
Old Smokey