Let $\mathbb{D}:=\{ x\in\mathbb{R}^2~:~\vert x\vert<1\}$ denote the open unit disk and let $\overline{\mathbb{D}}$ denote its closure.
For both questions assume that $A\subset\mathbb{D}$ and $B\subset\mathbb{D}$ are finite and disjoint sets.
Question 1:
Let $f:B\to\mathbb{D}\backslash A$ be injective. Can you extend $f$ to a homeomorphism $\tilde{f}:\overline{\mathbb{D}}\to\overline{\mathbb{D}}$ that is homotopic to the identity relative to $\partial\mathbb{D}\cup A$?
Question 2:
Let $\phi:\overline{\mathbb{D}}\to\overline{\mathbb{D}}$ be a homeomorphism that is homotopic to the identity relative to $\partial\mathbb{D}\cup A$ and that restricts to the identity on $B$, i.e. we have $\phi(b)=b$ for all $b\in B$. Is $\phi$ homotopic to the identity relative to $\partial\mathbb{D}\cup(A\cup B)$?