Does the operation of addition on the subspaces of $V$ have an additive identity? Which subspaces have additive inverses?

Additive identity is the zero subspace $\{0\}$.
For additive inverses if $U + W = \{0\}$ then $U = W = \{0\}$ because $U\subseteq U+W = \{0\}$ and likewise for $W$. Thus subspaces don't have additive inverses (unless they're zero themselves that is)