Can I use 1 resistor for both inputs of an RTL NOR gate?
No you cannot (in general) use just one resistor and short the inputs together, assuming that's what you are doing. 

If you connect (short) multiple RTL outputs together you'll have a so-called wired-OR circuit (actually it will be an AND circuit) and that will appear to work (at least until all those output resistors in parallel become too low), and the inverter will give you a NAND output. 

However, by connecting the outputs together they no longer have individual logic states and cannot be connected anywhere else and appear as the correct logic state. In other words the wired-OR (AND) function is occurring at the OUTPUTS of the gates that are connected together rather than inside the gate. 

You think you are seeing a NOR function because open-circuit is actually appearing as a high (1).