Vakil defines $\mathcal{O}_{\mathbb{P}_k^1}(n)$ as follows: it is Spec $k[x_1/x_0]$ on $U_0 = D(x_0)$, and Spec $k[x_0/x_1]$ on $U_1 = D(x_1)$, and the transition function from $U_0$ to $U_1$ is multiplication by $(x_0/x_1)^n$.Then he claims that from this definition it's obvious that $\mathcal{O}(n) = \mathcal{O}(1)^{\otimes n}$.
I'm not sure if I completely understand transition functions. Is it basically giving a glue along the open sets $D(x_0/x_1) \subset U_0$ and $D(x_1/x_0) \subset U_1$?
I don't see how $\mathcal{O}(n) = \mathcal{O}(1)^{\otimes n}$ follows easily from this transition function definition. Tensor product of sheaves is pretty mysterious to me because you have to sheafify.
Regarding 2, you can use that, if $F$ and $G$ are quasi-coherent sheaves, and $U$ is an affine open, then $(F \otimes G)(U) = F(U) \otimes G(U)$, where the latter tensor product is over $\mathscr{O}(U)$. This is also true of morphisms between quasi-coherent sheaves. This will let you compute the tensor product, and conclude the 'obvious' fact. (It's not obvious.)
See here for a reference: Why tensor product of two sheaves of modules is a sheaf of modules?
It's also in Ravi, around where he introduces QCoh, I think.