If the aircraft is truly incapable of taking off -- the locked controls case, let's say -- then the only choice involved is along the lines of, go off the end of the runway really really fast (you kept accelerating once past takeoff speed), or go off the end of the runway not so fast, with max braking, max reverse, full spoilers, etc. So if both engines just failed, it matters not how fast you are in relation to V1, you'd better start stopping, because you can't go flying -- at least, not for very long, and that certainly wouldn't end well either!
In the world of high-performance jets, the V1 speed is the speed at which you have determined you will commit to taking off. That's it. Before that speed, if I get a serious problem, I'll stop; above that speed, even with a serious problem (typically, along the lines of a fire or an engine failure), I'll continue the takeoff and deal with the problem in the air. And the performance data supports continuing the takeoff from that point even with one engine out.
So the statement -- which you'll see many, many places -- that "above V1 you can't abort & stop on the runway" isn't always accurate. If I have a powerful & light aircraft that needs 3,000' of runway to accelerate to takeoff speed, and another 3,000' to come to a stop if an abort is initiated at takeoff speed, and I'm taking off from a 12,000' long runway, then the speed that I'll use for the "V1" callout will be my takeoff (i.e. "rotation") speed, VR. But, could I successfully abort from a speed faster than that in this case? Absolutely.
Consider the case when my jet has a light load of passengers + fuel, and the identical aircraft taking off behind me has a heavy load. His VR speed, when he rotates up and starts to go flying, will be higher than mine because he has a heavier aircraft. So maybe he needs 4,000' to accelerate to his takeoff speed, and 4,500' feet to stop from that speed. So his VR is above mine, and his V1 matches his VR, so his V1 is higher than mine... well, if he can stop on the runway having initiated an abort at the higher speed, in my lighter aircraft I certainly could also initiate an abort at that higher speed and stop successfully.
We cap our V1 speed at VR, because it doesn't make sense, in a planning & performance standpoint, to consider rejecting the takeoff after you've started to rotate & go fly. But at any time up until that V1 speed, you aren't committed to flying yet... you can still abort.
As for timing, how quickly the Pilot Flying would have to initiate the abort after event (let's say the "bang" of an engine seizing -- a really violent engine failure), that's defined by the manufacturer and incorporated into the performance data. They test how long, in given conditions, it takes from the time the pilot starts the abort procedure, until the aircraft is stopped.
Then they build in assumptions (which they state in the performance data) about reaction time and so on. So if the test pilots, knowing what's coming, reach XXX knots & shut down an engine, they'll delay some time (perhaps 1 second) to say "Abort!", reflecting that everybody besides the test pilots has to process what just happened & will take a non-zero timespan to determine than an engine failure occurred.
Then, having stated "abort," they may delay one more second, and initiate the abort procedure -- perhaps one engine in idle (simulating, failed), one engine in reverse, full braking on the wheels, full spoilers. And when the aircraft stops, that distance is the data point.
Then they repeat that test at enough different conditions (gross weight, temperature, etc) so that they can confidently draw up performance charts so that I can tell, on "this" day with "these" conditions that I have, what is my accelerate-stop distance and speed -- what will it take for me to go from stopped, to takeoff speed, to stopped again, using all the given assumptions? If my VR speed is higher than this speed, then my V1 will be below my VR (and at, or perhaps slightly below, this accelerate-stop speed determined from the charts & the available runway). Or, if I can accelerate to a speed higher than my takeoff speed & still stop within the available runway, then my V1 probably matches my VR.
So to your (a) question above, yes, the odds of "successfully" aborting above V1 depend greatly on how far above V1 the pilot initiates the abort. If it's a half-second beyond, maybe the jet stops in the overrun, with no or minor damage, while 5 seconds beyond, the jet is through the overrun, the approach lights, the boundary fence, and into the highway just outside the airport. Or, maybe this was the day you only needed 6,000' to accelerate & stop again but you had 12,000', so despite the late abort everything turned out okay. Or maybe the runway has EMAS, and that ended up protecting the approach lights, fence, and highway.
As for your question (b), this is where people get creative, and sometimes a little bit aircraft-specific. A total electrical failure on the runway, at night with an overcast you'd be climbing into might be such a scenario. In the T-38, there was a specific combination of generator failure with an additional electrical failure that left you with a pretty ugly set of instrument outages, and it was a discussion topic of, do you take the jet into IMC in this condition, or do you abort late & trust the barrier to stop you. Maybe a massive armada of aircraft suddenly appearing at 200' with the certainty that going flying means a midair with one or more of them would qualify (like I said, getting creative here).
But for the most part, the aircraft designers have put lots of work into designing out as many of those sorts of scenarios as possible, so that you can safely go flying when an engine fails at or above V1.
For question (c), in an airline or military world at least, there would be an investigation into everything that happened, and did the PIC make an appropriate decision at the time, knowing what he knew and given the guidance that he had. If he aborted many seconds beyond V1 because he got a warning light on the cabin pressurization panel, the investigation will probably determine that he screwed up pretty seriously. Or, if he put the airplane into the EMAS rather than taking the total electrical failure airborne on the dark & stormy night, they could conclude that he made a really good decision.
Depending on how punitive the employer is and how badly he messed up, the pilot might be fired, or might get some (punitive) time off & some retraining, or there might simply be a debrief and some "lessons learned" for everyone, and he keeps on flying. In the case of a sufficiently egregious screw-up and an NTSB investigation, the FAA might pull his license so he's done flying altogether.