$$\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right| = 2,\,\theta \in (0,\,\pi )$$

$$ \Rightarrow {\left| {\widehat a + \widehat b + 2(\widehat a \times \widehat b)} \right|^2} = 4$$

$$ \Rightarrow {\left| {\widehat a} \right|^2} + {\left| {\widehat b} \right|^2} + 4{\left| {\widehat a \times \widehat b} \right|^2} + 2\widehat a\,.\,\widehat b = 4$$

$$\therefore$$ $$\cos \theta = \cos 2\theta $$

$$\therefore$$ $$\theta = {{2\pi } \over 3}$$

where $$\theta$$ is angle between $$\widehat a$$ and $$\widehat b$$.

$$\therefore$$ $$2\left| {\widehat a \times \widehat b} \right| = \sqrt 3 = \left| {\widehat a - \widehat b} \right|$$

(S1) is correct.

And projection of $$\widehat a$$ on $$(\widehat a + \widehat b) = \left| {{{\widehat a\,.\,(\widehat a + \widehat b)} \over {\left| {\widehat a + \widehat b} \right|}}} \right| = {1 \over 2}$$

(S2) is correct.