Homotopy classes of simple closed curves in punctured torus are in correspondence with those in torus

Question

I am reading the book "A Primer on Mapping class groups" by Farb and Margalit and I have two questions about homotopy classes of simple closed curves in $S_{1,1}$:

1. Why is it true that homotopy classes of simple closed curves in $S_{1,1}$ are in correspondence with those in $T^2$? (I have read this, but don't understand it.)

2. Let $\alpha,\beta$ be two simple closed curves in $S_{1,1}$ that intersect at one point. If we cut $S_{1,1}$ along $\alpha\cup\beta$, we obtain a once-punctured disk. Is the cut surface a union of disjoint disks if I consider $\alpha,\beta$ as the two generators in $\pi_1(T^2)$? How do we have the punctured disk?

Thanks for any help!