Hello Francois,
Thank you for your sustained help. We used your advice to progress on both methods but still hit roadblocks.
Volume approach:
We managed to open the strange output in MRIcron. When we tried to overlay the MNI template, we had an error saying that the files are not co-registered.
Surface approach: We did get confused with the two .gii, as you predicted. Once that was fixed, we tried to follow the ExportSpm12 instructions but we are under the impression that it doesn't apply exactly to our dataset. The instructions seem to be more focussed on stats analysis, whereas we merely want to see the surface. We don't have a "baseline" either in this condition. Given that, we were not able to reach the surface visualization.
We also tried to open the correct .gii in SUMA. It worked, but we didn't have the option to inflate the brain to match the previous figure. We will post on their forum to see if something can be done from that end.
Thanks again