Is the home team responsible for the setting of the boundary? Can they set the boundary in such a way that may benefit them? E.g., supposing they tend to hit the ball straight (or to cow corner), can they shorten the straight (cow corner) boundaries and lengthen the other boundaries? (And, if the boundary needs to be in a certain shape, can they shorten/lengthen it at will?)
The short answer is NO. Home team is not supposed to set the boundary as they wish. Some rules are applied for boundary setting as well. Even though the home team is responsible for setting up the ground.
As per ICC Standard Playing Conditions, the minimum and maximum size of the playing surface. Law 19.1 of ICC Test Match Playing Conditions states:
"The playing area shall be a minimum of 150 yards (137.16 metres) from boundary to boundary square of the pitch, with the shorter of the two square boundaries being a minimum 65 yards (59.43 metres). The straight boundary at both ends of the pitch shall be a minimum of 70 yards (64.00 metres). Distances shall be measured from the centre of the pitch to be used. In all cases the aim shall be to provide the largest playing area, subject to no boundary exceeding 90 yards (82.29 meters) from the centre of the pitch to be used. "
They can set the boundary as long as they follow the above rule. But the above rule doesn't give the freedom to reduce the boundary a lot. It should be a minimum of 65 yards and max of 90 yards. So the chances of setting a boundary as mentioned in question is very less.