To add my answer to the question of when to abort and when to keep going...

In theory, the V1 speed is the go/no-go speed. Any problem before V1 and you abort, any problem after V1 and you continue the take-off since at that point it takes less runway to continue than to stop. The other relevant speeds are Vr (rotate to nose up attitude) and V2 (should be passing 50' and climbing).

I say "in theory" because judgement does enter into the decision - you certainly wouldn't want to try taking off if both engines quit just after V1, you'd want to apply full braking regardless of runway left because the airplane would start slowing down and have no chance of getting airborne, never mind actually flying around for an approach. Also entering into the judgement call is runway remaining. V1 is calculated assuming that is just long enough to get airborne as long as everything is normal. For a relatively short flight like PHL-FLL (with a plane capable of flying transcon) there could be a pretty good amount of runway 27L still available at V1 and Vr.

The apparent fact that the plane got airborne makes it sound like something happened after Vr - wind gusts could cause airspeed fluctuations but with normal acceleration after Vr that wouldn't make much difference unless the gust was appreciably greater than the base wind. Another place judgement enters the picture - if you're getting 10-20 kt fluctuations in airspeed during the takeoff roll don't be in a big hurry meeting V1/Vr standards. Delay both some to allow for possible gusts. Better too much speed than too little once airborne.

What happened between approaching V1 and being airborne? We'll have to wait for the NTSB investigation for that. It certainly appears that whatever happened, it caused the crew to change their mind (in that instant they had) about the ability of the plane to continue flying.

Just an example - PI had a 727 taking off on what was 14 at IAH at the time (now 15L) - 12,000' long with a normal takeoff using less than half. They experienced a turbine failure in the #1 engine just after Vr and elected to just pull the power back and land. As it turned out, turbine blades from #1 penetrated #2 and that engine could have quit at any time. The hydraulic system that retracts the gear/flaps uses engine driven pumps on #1 & #2 so they could have lost the ability to retract gear/flaps at any time. The 727-200 can't continue to fly on one engine with the gear down - in fact, gear down is the commit point for an approach to landing. There's no go-around capacity with the gear down - the ibw remaining engine will only take you to the crash site if you attempt a go-around. So their decision, made in that split second, could have prevented an off-airport crash with loss of life. That's the judgement that is always required when something goes wrong.

Jim