Weak axiom of Revealed Preference application
Because WARP would imply that Barbara is chosen everywhere when Barbara is a choice in the set.


This is not true. Instead WARP would say that it is not possible for Andrew to be chosen whenever both Andrew and Barbara are in the choice set, which is different than your claim above.

Just because $b$ is chosen from the set $\{a,b\}$ doesn't mean that $b$ should always be chosen, e.g. from the set $\{b,c\}$. It just means that $b$ is revealed-preferred to $a$, but it doesn't mean that $b$ is preferred to all other possible alternatives (in particular, $c$). For example, the preference $c\succ b\succ a$ is perfectly consistent with the revealed choice pattern of the firm.

Write out the two conditions explicitly: 
$$
\begin{align}
C(\color{red}{\varnothing},\color{red}{\{a\}},\{b\},\color{red}{\{a,b\}})&amp;=\{b\}\tag{1}\\
C(\color{red}{\varnothing},\color{red}{\{b\}},\{c\},\color{red}{\{b,c\}})&amp;=\{c\}\tag{2}
\end{align}
$$
$(1)$ just says that whenever the four choices --- hire no one, hire $a$ only, hire $b$ only, hire both --- are present, the firm hires $b$ only. 
Likewise, $(2)$ says whenever the four choices --- hire no one, hire $b$ only, hire $c$ only, hire both --- are present, the firm hires $c$ only. 
These two are consistent with WARP (for $(1)$, the x in your quoted definition is $a$ and y is $b$; for $(2)$, x is $b$ and y is $c$). 

For restrictions on $C(2^{\{a,b,c\}})$, writing it out explicitly, we get 
$$
C(\color{red}{\varnothing},\color{red}{\{a\}},\color{red}{\{b\}},\color{black}{\{c\}},\color{red}{\{a,b\}},\color{black}{\{a,c\}},\color{red}{\{b,c\}},\color{black}{\{a,b,c\}})=\{\{a\},\{a,c\},\{a,b,c\}\}
$$
where the elements colored in red are revealed to be inferior according to $(1)$ and $(2)$.