...
Copper + Gold + Copper + Gold + Copper
0.5 µV/°C + 0.5 µV/°C + 0.5 µV/°C + 0.5 µV/°C + 0.5 µV/°C
You are perhaps forgetting polarity. Copper-Gold is 0.5uV/'C but Gold-Copper is -0.5uV/'C so you will get a good deal of cancellation. Also, the plating is intimate contact with (and so, the same temperature as) the underlying metal of the connector, so whether this is Copper, brass, Nickel etc. has a much greater consequence than the plating. This applies both to the mated contacts and the copper wire to connector junction (solder or crimp).
As you observed, the thermal emf reading tends to decrease with time as the connector temperatures equalize. Other things being equal, accumulated thermal emf the +ve and -ve connections should cancel as their temperatures become the same.
If I were to use copper all along I would straight away start getting, <0.3 µV/°C
Yes you should (and these should mostly cancel over time, as above). Note though that Copper-Copper Oxide is >500uV/'C. The Copper needs to be freshly clean, otherwise the thermal emfs can be huge and unequal.
It might be worth getting one of the guys in your lab to give you a more detailed refresher, as this can be quite a significant factor for low level, or long scale instrument calibration uncertainties.