Is the Euler characteristic of $\mathbb{R}^n$ $1$ or $(-1)^n$?

Your definition of Euler characteristic does not agree with the notes you link to. Nicolaecu's notes talk about compactly supported cohomology groups, and it is true (as he calculates, that $H_c^k(R^n) \cong 0$ if $k\neq n$ and $\cong R$ if $k=n$, hence $\chi(R^n)=(-1)^n$ with his definition of Euler characteristic (there is also another notion (and more commonly used) of Euler characteristic, which is the one you define, but it is different from the compactly supported one)

The ball $B^n$ is not homeomorphic to the disjoint union of the boundary sphere and its interior. So the calculation $\chi(\mathbb{R}^n) = \chi(pt) = 1$ is the correct one.