Does this equation - involving calcium carbonate and a lead (II) cation - balance? $$\ce{CaCO3 + 2H2O + Pb^{2+} -&gt; Pb(CO3) + Ca(OH)2 + H2}$$ 

Does this equation balance? 
On the right you have lead with 2+ charge, whereas on the right its completely neutral. So my question is do you need to balance the charges in order to make the equation correct? 

If that equation is wrong, what is the right equation? 

I need to show that hydrogen gas is formed (or some other gas but I don't know what else it can be), that the solution will become more basic (proven with $\ce{Ca(OH)2}$), and that the lead has to replace the calcium.
Unfortunately your equation does not add up, let me briefly explain why.

Calcium carbonate is a not very soluble salt. There is ionic binding between the calcium cation and the molecular carbonate anion. If you dissolve it in water you, the following equilibrium will establish: $$\ce{CaCO3~(s) &lt;=&gt;[\ce{H2O}] Ca^{2+}~(aq) + CO3^{2-}~(aq)}$$

Lead carbonate is also a salt, it is even less soluble. Only a little bit will dissolve according to the following equilibrium:$$\ce{PbCO3~(s) &lt;=&gt;[\ce{H2O}] Pb^{2+}~(aq) + CO3^{2-}~(aq)}$$

Calcium hydroxide is also a salt. It is a little bit more soluble than calcium carbonate. The following equilibrium is established when put in water:$$\ce{Ca(OH)2~(s) &lt;=&gt;[\ce{H2O}] Ca^{2+}~(aq) + 2{}^{-}OH~(aq)}$$

You write the following:


  $$\ce{CaCO3 + 2H2O + Pb^{2+} -&gt; Pb(CO3) + Ca(OH)2 + H2}$$


So in principle you have $\ce{Ca^{2+}}$, $\ce{Pb^{2+}}$ and $\ce{CO3^{2-}}$ on both sides of the equation. What is left is not a balanced equation, since there are two electrons missing:
$$\ce{2H2O ~\nrightarrow 2{}^{-}OH + H2}$$

You need a source for the electrons, one way of balancing the equation is
$$\ce{CaCO3 + 2H2O + Pb -&gt; Pb(CO3) + Ca(OH)2 + H2}$$

Given the insolubilities of all the species involved, it might be hard to see any reaction at all. And I have not checked if the redox reaction is favourable, with potentials.