Three contour and winding-number calculations

Evaluate or prove the following.

1. \[ \int_{|z|=2}\frac{e^z}{(1+z)^2}\,dz. \]

2. For \(\gamma(t)=2\cos t+i\sin t\), \(0\le t\le2\pi\), evaluate \[ \int_\gamma\frac{dz}{z^2+2iz}. \]

3. If \(\gamma\) is a piecewise smooth closed path, verify \[ n(\overline\gamma,\overline z)=-n(\gamma,z). \]