What happens to the kinetic energy of a snowball as it rolls across the lawn and gains mass?

Well, there's a lot of friction going on there, so the snowball gradually
loses kinetic energy just from bouncing and plowing through the snow
on the ground.

But I don't think you're asking about that.  I think you're ignoring that
for the moment, and asking how its kinetic energy changes as its
mass increases.  We know that

                               Kinetic Energy = (1/2) (mass) (speed²)

and THAT seems to say that more mass means more kinetic energy. 
So maybe the snowball's kinetic energy increases as it picks up
more mass.

Don't you believe it !

Remember:  Energy always has to come from somewhere ... a motor,
a jet, a push, gravity ... something !  It doesn't just appear out of thin air.
If the snowball were rolling down hill, then it could get more kinetic energy
from gravity.  But if it's rolling on level ground, then it can never have any
more kinetic energy than you gave it when you pushed it and let it go.

If snow or leaves stick to it and its mass increases, then its speed must
decrease, in order to keep the same kinetic energy.

Domain eukarya; kingdom plantae should be the answer.

I’m not sure about this question