Does the equation x2 - 4x + y2 = -3 intersect the y-axis?

No.  The y-intercept is the value of y when x=0, hence it intersects the y-axis when x=0.  In this case:

0^2-4(0)+y^2=-3

y^2=-3

y=√-3

Since there is no real square root of a negative value, the curve does not intersect the y-axis