Question

What is the electric potential and electric field due to a point charge at the point where the point charge is kept(i.e. when r=0)?

Answer 1

R can never reaches zero, because it would require a point charge to have dimension i.e. a definite position and velocity (or momentum).That is impossible because something as small as a point charge cannot simultaneously have a definite position and momentum.

Consequently electric field and potential cannot have the value in the immediate vicinity of a point charge.