the position of the system, in principle should not affect, as you say the Coulomb potential is a function of |r-r'|.
Anyway note that your calculation, due to the different position in space have a different number of symmetries.
Moreover it seems to me that something happened in the ground state calculation as you have different gap at gamma:
report_0
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[X] Direct Gap : 0.604562 [eV]
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[X] Direct Gap : 1.551236 [eV]
The box cutoff, anyway i general has some problem of accuracy, and it will be soon dismissed and substituted with a more accurate method.
If you want in the meantime you can give a try to use the "ws cutoff":
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CUTGeo= "ws z"
Unfortunately, the k point convergence is something that has to be checked. GW band gap and absorption spectrum in BSE usually require grid larger than the one needed to converge the ground state density, but this depends of the bands profile, exciton localisation etc...I have considered the k point sampling that I used for the optimization after a convergence study on the final energy and I am calculating the gap on Gamma. Is it good this way or should I do different Abinit runs with different k-points meshes to check the GW gap? The last option could take a lot of time.
Best,
Daniele