For Question 1, notice first that you can color the set of positive real numbers with two colors so that $x$ and $2x$ always have different colors: Color $x$ according to the parity of $\lfloor\log_2x\rfloor$. Now assign to each circle the color that you just gave its radius. Since the circumcircle of an equilateral triangle has twice the radius of the incircle, these circles will always have different colors.

Question 2 seems to have a negative answer for a similar reason.

For Question 3, the answer is affirmative. Given any coloring with two colors, consider three concentric circles with radii $1,\sqrt2,2$. The first and second are the incircle and the circumcircle of a square. The second and third are the incircle and the circumcircle of another square. The first and third are the incircle and the circumcircle of an equilateral triangle. Since two of the three circles must have the same color, at least one of the squares or the equilateral triangle has its incircle and the circumcircle colored the same.

necessary? $\endgroup$ – Gerry Myerson Jul 5 '18 at 22:56