The calculation of $\dim(U + V + W)$
Solution 1:
I don't think you can do better than this: $$\begin{align*} \dim(U +V + W) &= \dim((U +V) + W) \\ &= \dim(U +V) + \dim W - \dim((U+V)\cap W) \\ &= \dim U + \dim V - \dim (U \cap V) + \dim W - \dim((U+V)\cap W) \end{align*}$$ Now you're stuck with $\dim((U+V)\cap W) $, for which there does not seem to be a simple formula.
(BTW, see also https://mathoverflow.net/questions/17740/is-there-a-version-of-inclusion-exclusion-for-vector-spaces but you probably know about this already.)