1) Let $C$ be a contour beginning and ending at 1. Suppose that $f(z)$ is analytic on $C$. Then is it true that the contour integral of $f$ around $C$ is 0?
This looks to be true by Cauchy's theorem but I'm not sure about the interior points part of the hypothesis
2) Let $C$ be a contour beginning and ending at 1 and containing numbers whose real part is positive. Then is it true that the integral of $\frac{1}{z^2+z}$ around $C$ is zero.
If all you have is that $f$ is differentiable on $C$, then $f$ could have singularities elsewhere, each of which would affect the integral of $f$ around $C$ if the winding number of $C$ around it is non-zero. Have you integrated $z \mapsto z^{-1}$ around the unit circle before? That is the most important integral because it tells you essentially what happens at any singularity since the integral is $0$ for all other integral powers.
Also, as you might have noticed, the conditions for Cauchy-Goursat theorem to hold is that the curve must be closed and rectifiable, and the function must be differentiable on a simply connected set that includes the curve. Can you see whether any of the conditions fail to hold for (2)?