On the RR 777 and the RR A330 the FADEC won't put the fuel in until the EGT drops below 100c during Auto start. It motors the Engine at Max Motoring until the temp drops.
As to the Engine N2 either accelerating fast or slow, that would depend on bleed pressure in the duct at the time wouldn't it? Have a look at the Duct pressure when it happens.
As to the APU N2 varying with speed, that would be down to the APU Battery condition?