Yes, or a thermistor, or even put the ADC close to the terminal block and use the temperature sensor which a lot of them have on-chip (e.g. ADS1118).
Well, yes, exactly. That is very hard to avoid if there is any, ahem, electronics in the box :)
I am seeing 2-3C delta T between the PCB, 5mm away from the terminal block, and the actual cold junction which is at the screw terminals. I will try to characterise this, as I suspect it is fairly constant (within 1C or so, and most people using thermocouples will never notice anything better, in typical tc apps).