Is the induced electric field due to time varying magnetic flux always circular? According to Faraday's law, changing magnetic flux induces an electric field. Is that electric field always circular? 
 A: Your question is like asking do charge particles always produce a spherically symmetric electric field?  The answer is yes and no.  A single electric charge, like an electron, will produce a spherically symmetric electric field, but if I have a collection of charged particles, then the shape of the resultant electric field will depend upon the geometry of the distribution of the charged particles.
Let us examine Faraday's law:
$$\nabla \times E = - \frac{\partial B}{\partial t}.$$
Using the Stoke's theorem we arrive at:
$$\int E . ds = - \frac{\partial \int\int B. dA}{\partial t}. = - \frac{\partial magnetic flux}{\partial t}$$
Let us suppose that we have cylindrical symmetric magnetic field heading from North to South and varying in intensity over time, thus giving a non-zero value for $\frac{\partial magnetic flux}{\partial t}$.  Now if we choose the line integral to be a circle around this magnetic field, due to symmetry, $E$ must be the same around all points of this line, and thus we could say that this magnetic field produced a circular electric field.
However, this example is a non-realistic case of a cylindrically symmetric magnetic field.  In reality, the distribution of the magnetic field won't be so symmetric, thus causing non circular electric fields (although still closed loops of electric fields).
