Which parameters in datasheet give information about a BJT's step response quality?
It is usually capacitance that causes problems - and it's usually the miller capacitance that is the main culprit. The miller capacitance is internally from collector to base and if the base is driven from a high impedance (10k in your diagram - really quite high to expect decent performance) the miller capacitance forms a low pass filter with the input resistor.

If miller C is 10pF and input resistance is 10kohm then these two components form a low pass filter with a cut-off at: -

\$f_C = \dfrac{1}{2\pi R C}\$ = 1.59 MHz.

Try reducing the 10k ohm input resistor to something more sensible like 330 ohms and see the difference. I simulated a BC817 as you have drawn it and it took about 2.3us to reach 11 volts when switched off. With a 1k input resistor it took less 0.33us. With 330 ohms it took about 0.25us.

If I lowered the collector load to 1k and kept the 330 ohm input resistor it took about 0.09us.

The lowering of these resistors all serve to reduce the effect of miller capacitance (amongst other things).