You might consider an analogy of a bottle of compressed air. If you open the bottle, air escapes into the atmosphere until the pressure inside and out is "atmospheric pressure". The atmosphere could be considered the "infinite reservoir " here.
If you were inside a sealed container - say a submarine - when you did this, you'd find the same: air would escape until the pressure inside and outside the bottle was equal to the submarine's internal air pressue.
In either case the flow of air would increase the pressure outside by a tiny amount. For the world atmosphere I think this would be negligible. For the submarine, perhaps it were a very large tank (rather than the pop bottle or beer can I'd thought about) maybe the pressure change would be detectable?
You are correct that your charge goes somewhere, and it changes the potential of the "sink" : just as the object is not completely discharged (in general) but brought to the same potential as the sink.
Whether the ground becomes negatively charged by discharging your object, would depend on it's original state of charge. If the Earth were already charged with positive* megacoulombs, then a few picocoulombs of electrons would not make it become negative.
Edit: on looking into the charge on the Earth, I can't find anything on which 97% of scientists agree, but more seem to say our charge is negative. The magnitude maybe of the order of 0.6 MCb.