(Unfortunately I am not able to access the video at the moment).
Your question is difficult only because you are picking bits and pieces out of the Standards.
Easiest summary might be
(a) expected climb performance should be not less than gross minima (and, in general, very marginally better than gross depending on the aircraft and engine condition)
(b) the AFM will provide a bunch of data for net degraded conditions
(c) obstacles must be below the net flight path as constructed iaw the AFM for the day. If the sums show this is not the case, then the take off parameters need to be varied to achieve the requirement .. either by reducing weight and/or increasing the speed schedule depending on what and where the problem is. This will result in climb gradients greater than the minima specified in the Standards.
For your specific questions
(1) second segment minimum gradient for a twin is 2.4% gross (1.6% net)
(2) FAR 25 has nothing to do with obstacle clearance
FAR 25 provides the tools to do the sums to miss the rocky bits.
Operational requirements (and commonsense) suggests that obstacles must be below the aircraft flight path. For the calculations in respect of obstacles, the relevant flight path is the net flight path.
Per 121.189(a) the AFM requirements are the starting point.
(3) FAR 121.189 ... the net takeoff flight path for the 2nd segment would only be required to be 1.6%, and that gradient would be required to clear the obstacles by 35'?
You're mixing things up a bit here.
The minimum net second segment flight path is 1.6% which provides a limiting weight.
Note that the limiting weight for the takeoff is the most limiting weight of the various limiting weights which have to be checked. That is, second segment limits may/may not be what limits the RTOW for the particular takeoff.
If the particular runway on the particular day is suited to the performance obtained at that weight .. fine .. otherwise you reduce the weight and/or increase the speed schedule to improve the gradient and, hence, obstacle clearance. That is the runway parameters provide additional input into the RTOW calculation.
Be careful not to consider climb limits in isolation to the other stuff which needs to be considered.
I had always been under the impression
(Caveat .. for twin engined aeroplanes). OK except that the minimum gradient requirement for second segment is 2.4% gross/1.6% net.
(4) WAT .. chart ensures compliance with all FAR 25 performance requirements for takeoff.
Not the case. The WAT data gives you the weight restrictions to meet the various minimum climb requirements
It is based on the most restrictive standard under FAR 25.
Not quite sure what you mean here ?