Electronic – How to use the “Switching characteristics” parameters in the BJT’s datasheet

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).

BC817 is a particular type of transistor, so any variations between suppliers will be minimal.

The requirements for this application are minimal. Since we can assume the battery voltage is less than 20V, the Vceo rating needs only to be 20V.

As to the base resistor- it sets the base current- the base-emitter junction behaves like a diode (around 0.7V drop). Most any ordinary inexpensive transistor you can find will have a beta of more than 40. We don't need the transistor to saturate hard in this case, we only need a voltage of more than (say) 4.5V across the 100K resistor, which implies a minimum collector current of 45uA. If the beta is 40, then the base needs to see about 1.1uA. If we assume the minimum supply voltage for our GPIO is 1.8V * 0.95 and the maximum Vbe is 0.8V, then the maximum base resistor is about 800K, so 2K seems pretty safe, even with a really low voltage drive (20K would be fine too). The lower limit on the base resistor is determined by how much current you want to draw from the GPIO.

The leakage should be well under 10uA under all operating conditions to avoid turning the MOSFET on (only potentially an issue at very high temperatures with modern transistors).

It should be NPN type. Avoid RF transistors and exotic SiGe types, they would probably not misbehave but they're of no help, and may be rated for unusually low voltages.

Pretty much any general purpose switching transistor will work here. 2N3904, 2N4401, BC817, 2SC1815, etc. etc. You can search for the most common type and/or least expensive and it will probably meet all the requirements when you look at the datasheet.