The base-emitter path is not
quite a short, but it's close enough to be bad to short out. A quick run through LTspice says 5V applied directly to the base of a 2N2222 gives about 400mA even with no collector current, and a power dissipation in the transistor of 2W. The Fairchild PN2222A datasheet specifies a maximum power of 0.625W. It does not give a maximum base current, but the NXP datasheet for the original metal can 2N2222 says 200mA. So not quite a short, but highly in excess of the absolute maxima. The NXP 74HC4050 datasheet specifies a maximum output current of ±25mA, so 400mA is hugely in excess of that as well.
The datasheet will also have plots showing how the values change with different conditions. The Fairchild PN2222A sheet says that with V
CE = 12V, h
FE becomes about 1.05 times the specified value, and with V
CE = 20V, it becomes about 1.1 times the specified. It will also depend on collector current, of course. See the plots titled "Common Emitter Characteristics" on the last page,
http://www.fairchildsemi.com/ds/PN/PN2222A.pdf.
The pulldown resistor on the MOSFET will determine how long it takes to shut off, if the gate is
only driven high and just allowed to sink low by itself. This is very approximate, but it will take something on the order of RC seconds, where R is the pulldown resistance and C is the gate capacitance (check the datasheet). If you're not sure, feel free to include a pulldown as it will rarely interfere with proper functioning. If the gate is driven by a push-pull output (one that can sink and source), it should not need one. I wouldn't worry too much about the physical distance between them as long as it's always solidly connected.