So then, all the ions just disassociate and drift about freely in solution, and when they form less soluble combinations, that's the product?
Yes.
So what about Ca(OEt)2 + 2 NaOH --> 2 NaOEt + Ca(OH)2 ? Same deal?
Yes.
So then, in that case, couldn't a person, say, produce aluminum ethoxide by refluxing aluminum with ethanol, then react it with NaOH to yield NaOEt? Since it would seem that aluminum hydroxide is rather insoluble in anything much.
Theorethically, yes - but Aluminum is a somewhat special case, as it may form sodium aluminates with varying ethoxide and hydroxide groups to each molecule, and if you have ever tried to filter aluminum hydroxide from anything, then you know this isn't a practical idea.