Why does bubbling carbon dioxide through calcium hydroxide result in a precipitate?
The important thing to consider is the chemistry of carbon dioxide in water, forming carbonic acid. Note also that calcium hydroxide is at edge between being a strong or a weak base.

$$\begin{align}\
\ce{CO2 ^ + H2O &amp;&lt;=&gt; CO2 . H2O} \tag{1} \\
\ce{CO2 . H2O &amp;&lt;=&gt; H2CO3} \tag{2} \\
\ce{H2CO3 &amp;&lt;=&gt; H+ + HCO3^-} \tag{3} \\
\ce{HCO3^- &amp;&lt;=&gt;  H+  + CO3^2- } \tag{4} \\
\ce{ Ca(OH)2 &amp;&lt;=&gt; CaOH+ + OH- } \tag{5} \\
\ce{CaOH+ &amp;&lt;=&gt; Ca^2+ + OH-} \tag{6} \\
\ce{Ca^2+ + CO3^2-  &amp;&lt;=&gt; CaCO3 v} \tag{7}
\end{align}$$

Yes, bonds in $\ce{CO3^2-}$ are equivalent, with strength and length somewhere between a single and a double bond. It is similar as  for $\ce{SO4^2-}$, $\ce{SO3^2-}$, $\ce{NO3^-}$, $\ce{ClO4^-}$, with sharing the "doublebondness" within the former two, and sharing the charge in all of them.