What does 25°C mean on a transistor data sheet?
It's the junction temperature \$T_J\$, and yes, they would typically take such a measurement in microseconds before the junction temperature changes. 

The upward slope of at high base and emitter currents is due to resistive (non-ideal) behavior that is temperature-sensitive, or due to the reduction in beta at high current (so the base current is higher for a given collector current than if the beta was constant). Note that the beta reduction is temperature-sensitive.