Single elements of subspaces For two subspaces to be a direct sum of some vector space, must they each have no elements in common? 
E.g. Suppose U is the subspace of $F^3$  of those vectors whose last coordinates equals 0, and W is the subspace of $F^3$ of those vectors whose first two coordinates equal 0.
U and W would be a subspace correct? If, however, W is the subspace of $F^3$ of those vectors whose first coordinate equals 0 and U remains the same; would the sum of these subspaces be a direct subspace?
 A: Your reasoning is correct, the definition of $\textbf{direct sum}$ is the following:
A vector space $\textsf{V}$ is called the direct sum of $\textsf{W}_1$ and $\textsf{W}_2$ if $\textsf{W}_1$ and $\textsf{W}_2$ are subspaces of $\textsf{V}$ such that $\textsf{W}_1\cap \textsf{W}_2=\{0_V\}$ and $\textsf{W}_1+\textsf{W}_2=\textsf{V}$ (the last is the same as saying that for each $v\in\textsf{V}$ we have $v=w_1+w_2$, where $w_1\in\textsf{W}_1$ and $w_2\in\textsf{W}_2$).
The direct sum is denoted by $\textsf{V}=\textsf{W}_1\oplus\textsf{W}_2$. 
In your example:
$$\textsf{U}=\{(a,b,0):\, a,b\in F\}$$
$$\textsf{W}=\{(0,0,c):\, c\in F\}$$
Hence, any vector $(a,b,c)\in \textsf{F}^3$ satisfies that
$$(a,b,c)=(a,b,0)+(0,0,c)\in\textsf{U}+\textsf{W}$$
and the most important part: $\textsf{U}\cap\textsf{W}=\{(0,0,0)\}$. Thus, we can say that $\textsf{F}^3=\textsf{U}\oplus\textsf{W}$.
Added: Another definition (but equivalent to the one given previously) is that for any $v$ in $\textsf V$ this can be written in a $\textbf{unique}$ way as $v=w_1+w_2$, where $w_1\in\textsf{W}_1$ and $w_2\in\textsf{W}_2$.
A: No. Basically the only criteria for talking about (internal) direct sum of two subspaces $U,W\ \subseteq V$ is that they are disjoint, i.e. $U\cap W=\{0\}$, and then $U\oplus W=U+W$. 
In your first example this condition is satisfied, however $U\cap W=\{(0,a,0):a\in F\}$ in your second example, so $U+W$ is not the direct sum of $U$ and $W$. 
