A CW complex may have any point as a $0$-cell?

Question

I have two questions about CW complexes.

1. Let $X$ be any CW complex, and let $x$ be any point in $X$. Then may we assume that $x$ is always a $0$-cell of $X$? More precisely, is there a CW pair $(Y,y)$ (i.e., $y$ is a $0$-cell of $Y$) homeomorphic to $(X,x)$, or more generally homotopy equivalent to $(X,x) ~\text{rel} ~x$?

2. By Whitehead's theorem, a weak homotopy equivalence $X \to Y$ between CW complexes is a homotopy equivalence. Now suppose $f:(X,A)\to (Y,B)$ is a weak homotopy equivalence between CW pairs, i.e., $f$ induces isomorphisms $\pi_n(X)\to \pi_n(Y)$ and $\pi_n(A)\to \pi_n(B)$ for all $n$. Then we know that $f$ is a homotopy equivalence. However, is it true that $f$ is a homotopy equivalence rel $A$?

Thanks in advance