I fly B737s and have never heard of a lot of the things that you guys are talking about. LOL
Speeds used by the pilots on takeoff are:
V1 - Takeoff Decision Speed
Vr - Rotation Speed
V2 - Takeoff Safety Speed
Flaps are retracted in increments when reaching the minimum maneuvering speed for the current flap selection and accelerating.
A low-speed call-out is made to cross check airspeed indications and define the beginning of the high-speed reject regime. United uses 100kts, some airlines use 80kts.
I have no idea what happened on this particular flight but the description sounds like the engine "torched". That happens sometimes during an engine start, when the engine first lights off when some of the fuel has traveled into the exhaust section before igniting, and is unlikely to causes damage. Rarely happens to an engine that is already running unless there is a compressor stall but that would have been quite obvious to the crew.