Hi Federico,
You are using the LATTICE cards in the correct way. However it seems like you included the STOP card before defining the lattice ROT-DEFi and it is therefore ignored. Moving the STOP card after this should fix the geometry issues. You also did not set the (optional) continuation card for setting the geometry debugging step sizes, this is a line Fluka tries to read but does not manage (this needs to be fixed).
To my knowledge the lattice transformation will not affect the volumetric sources in the same way as is done for the geometry. The way forward would be to use the source.f routine (13.2.20} source.f: user-written source | FLUKA) as discussed in
User rotuine for multiple locations of a source - #4 by horvathd and
Zero result after biasing
There you define the same transformations as you impose on the different lattice cells.
Hope this is helpful, cheers,
Andreas