Reaction between sodium carbonate and water
The three reasons you propose are technically true, but they are not the reason why this is not the process used to produce sodium hydroxide.
But let me first clarify some points (and rant about a bit).

Hydrolysis is a reaction where a compound is split whilst incorporating a molecule of water. The reaction :
$$\mathrm{Na_2CO_3 + H_2O \rightarrow 2 NaOH + CO_2}$$
is not a hydrolysis. It is classified as an acido-basic reaction ($\mathrm{H^{+}}$ exchange) followed by a dehydration reaction. A dehydration is kind of the opposite of an hydrolysis (the true opposite would be called hydration), because is involves the loss of water. The transformation decomposes according to (forgetting about sodium ions):
$$\mathrm{{CO_3}^{2-} + H_2O \rightarrow {HCO_3}^- + HO^-} \\
\mathrm{{HCO_3}^{-} + H_2O \rightarrow {H_2CO_3} + HO^-} \\
\mathrm{H_2CO_3 \rightarrow CO_2 + H_2O }$$
$\mathrm{H_2CO_3}$ is carbonic acid. Carbon dioxide is its anhydride.
The overall transformation actually consumes two molecules of water as acid, and one molecule of water is produced upon dehydration of carbonic acid. But this does not appear in the simplified equation above.

This reaction is actually not thermodynamically favorable ; the reverse transformation is the one that is spontaneous. What drives it here is the fact that the carbon dioxide is gaseous and is driven off the water, into the atmosphere. Unless you work under a $\mathrm{CO_2}$-free atmosphere, it would not be possible to produce sodium hydroxide not contaminated with carbonate.

Now, why is this not used industrially ? The answer lies in the cost of the processes : investments, availability of reactants, price of energy, etc.

Calcinating sodium carbonate at more than 100°C actually gives off sodium oxide ($\mathrm{Na_2O}$), but it also releases $\mathrm{CO_2}$ in the atmosphere, and you have to obtain sodium carbonate first. This is no commonplace mineral. Most of it is actually produced by the Solvay process, which converts two plentiful materials, namely calcium carbonate and sodium chloride into calcium chloride and sodium carbonate :
$$\mathrm{2\,NaCl + CaCO_3 \rightarrow Na_2CO_3 + CaCl_2}$$

The chloralkali process produces sodium hydroxide (and chlorine) directly from sea salt, and the  products are hardly contaminated.