$\mathbf{Problem \,2.}$ Consider the $2$-manifold in $\Bbb R^3$ given by $$x^2+y^2+z^2=1,\qquad z\ge 0.$$ Orient $M$ such that $\alpha$ in the Equation $(2)$ belongs to the orientation, and give $\partial M$ the induced orientation.
$$\alpha(u,v)=\left(u,v,\sqrt{1-u^2-v^2}\right),\qquad u^2+v^2\lt 1$$
I think I'm completely lost here, what does orienting a manifold entail once I have coordinate patch?
Orienting your manifold $M$ means picking one of two possible directions for the unit normal vector to $M$. In your example, we have a 2-manifold that's the surface of a sphere in $\mathbb R^3$; it is the image of the function $\alpha(u, v) = (u, v, \sqrt{1 - u^2 - v^2})$, i.e., $\alpha$ is a parameterization of $M$. You could pick $\hat n(u,v) = - \frac{\nabla \alpha(u,v)}{\|\nabla \alpha(u,v)\|}$ or $\hat n(u,v) = \frac{\nabla \alpha(u,v)}{\|\nabla \alpha(u,v)\|}$ to be the unit normal vector (in $\mathbb R^3$) to $M$ at $\alpha(u,v) \in M$.