Yes, see. A bit terse but yes.
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GPS measure their coordinates in the ECEF (Earth Centered Earth Fixed) reference. That is with respect to Earth center of mass. That is natural since satellites orbit around that point.
With the help of a ellipsoid (part of the WSG84 system) and a Geoid one can get the z coordinate but lon,lat are obtained by geometry. UTM (and all other cartographic) projection is just that, a projection and as such it implies deformations. So, no, UTM is not the best referencing system when working with GPS data. But it can be good enough.
No, you can’t. Neither in GMT nor in any other software. Each zone (or graticule as is your case) is a different referencing system.
You have to convert your points into a common referencing system. Lon,lat, Transverse Mercator, Lambert (which the French like very much), etc… and then plot them.