Here some inline answers:
As I understand, v.overlay module can be used for polygons overlapping
Polygon clipping, union processes can be done. There are
many algorithms for these operations.
Which algorithm is used in GRASS for overlapping precesses?
GRASS GIS relies on the GEOS library for these operations: https://trac.osgeo.org/geos/
Technical details are found here.
Is it possible to parrallelize v.overlay module codes?
Probably yes, but liely it needs to be done in the GEOS library.
Why v.overlay module supports with two maps? Can I use it for 3 or more maps at the same time?
More than two overlaying geometry layers are not supported at time. You may open a feature request at https://trac.osgeo.org/grass/newticket (please select type: "enhancement" and component "vector", etc).