If you reject the take off prior to V1 you will stop within the accelerate/top distance available.
a bunch of assumptions is behind that sort of statement. Possibly better to talk about "generally" and, "if everything goes according to Hoyle" rather than making black and white statements.
the stopway .. may contain frangible obstacles which could cause minor damage to the underside of the aircraft.
probably not the case - perhaps you are thinking of clearway ? .. which is not specifically related to ASDA.
After V1 you will continue to accelerate to Vr
as previously observed .. it depends
become airborne at the end of the Take Off Run available.
definitely,
not the case. Depending on the certification rules, liftoff is scheduled to achieve either a minimum of a half or a third of the airborne distance to screen over the declared TORA and that only if TORR is the critical takeoff case for the day.
and clear of the limiting obstacle
separate calculation .. while often the TODA is predicated on obstacle clear surfaces, that is not a necessary condition.
still 35 feet clear of the limiting obstace
afraid that is nonsense. The net (straight ahead) surface can be sitting just above (ie 35ft) any critical obstacles .. but the aircraft will be above that surface with the delta increasing as the distance from the runway increases.
if necessary out to 15 kms.
generally out as far as it takes depending on the obstruction profiles. You may be confusing the usual Type A chart survey length ?
Oh goody, a chance to display my ignorance...
you too ? ... good health, sir.
In the real world it's quite possible to neither firewall the engines due to some misplaced fear of mountainous terrain nor risk damage to the frangibles if you have to perform an RTO.
a practical and pragmatic point of view.
And good fun it is too. 
especially empty on a really cold morning .. not often done, mind you, but clears out the mental cobwebs on the odd occasion.