If one looks carefully one notices a connector in the base circuit of the transistor in question. The resistor R1 in the base circuit isn't redundant, it is good engineering. If the connector is disconnected without the resistor the base bias is undefined. with it the base bias is defined for all conditions.
Not really, an NPN with open base will only conduct a small leakage current of nA to uA or so - not enough to switch a relay,
so is defined enough I reckon.
I suspect the board was designed to be compatible with either an NPN transistor or an n-channel MOSFET, and the pull-down
resistor is really there for the latter case - it causes no harm or benefit for an NPN transistor.